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Introduction to Probability
and Statistical Inference

This Page Intentionally Left Blank

Introduction to Probability
and Statistical Inference
George Roussas
University of California, Davis
Amsterdam Boston London New York Oxford Paris
San Diego San Francisco Singapore Sydney Tokyo

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This book is printed on acid-free paper.
∞
Copyright 2003, Elsevier Science (USA)
All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any
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Library of Congress Control Number: 2002110812
International Standard Book Number: 0-12-599020-0
PRINTED IN THE UNITED STATES OF AMERICA
02 03 04 05 06 9 8 7 6 5 4 3 2 1

To my wife and sons,
and the unforgettable Beowulf

Contents
Preface xi
1 SOME MOTIVATING EXAMPLES AND SOME
FUNDAMENTAL CONCEPTS 1
1.1 Some Motivating Examples 1
1.2 Some Fundamental Concepts 8
1.3 Random Variables 19
2 THE CONCEPT OF PROBABILITY AND BASIC RESULTS 23
2.1 Deﬁnition of Probability and Some Basic Results 24
2.2 Distribution of a Random Variable 33
2.3 Conditional Probability and Related Results 41
2.4 Independent Events and Related Results 51
2.5 Basic Concepts and Results in Counting 59
3 NUMERICAL CHARACTERISTICS OF A RANDOM
VARIABLE, SOME SPECIAL RANDOM VARIABLES 68
3.1 Expectation, Variance, and Moment Generating Function
of a Random Variable 68
3.2 Some Probability Inequalities 77
3.3 Some Special Random Variables 79
3.4 Median and Mode of a Random Variable 102
4 JOINT AND CONDITIONAL P.D.F.’S, CONDITIONAL
EXPECTATION AND VARIANCE, MOMENT
GENERATING FUNCTION, COVARIANCE,
AND CORRELATION COEFFICIENT 109
4.1 Joint d.f. and Joint p.d.f. of Two Random Variables 110
4.2 Marginal and Conditional p.d.f.’s, Conditional
Expectation and Variance 117
4.3 Expectation of a Function of Two r.v.’s, Joint
and Marginal m.g.f.’s, Covariance, and Correlation
Coefﬁcient 126
4.4 Some Generalizations to k Random Variables 137
4.5 The Multinomial, the Bivariate Normal, and the
Multivariate Normal Distributions 139
vii

viii Contents
5 INDEPENDENCE OF RANDOM VARIABLES
AND SOME APPLICATIONS 150
5.1 Independence of Random Variables and Criteria
of Independence 150
5.2 The Reproductive Property of Certain Distributions 159
6 TRANSFORMATION OF RANDOM VARIABLES 168
6.1 Transforming a Single Random Variable 168
6.2 Transforming Two or More Random Variables 173
6.3 Linear Transformations 185
6.4 The Probability Integral Transform 192
6.5 Order Statistics 193
7 SOME MODES OF CONVERGENCE
OF RANDOM VARIABLES, APPLICATIONS 202
7.1 Convergence in Distribution or in Probability and Their
Relationship 202
7.2 Some Applications of Convergence in Distribution:
The Weak Law of Large Numbers and the Central
Limit Theorem 208
7.3 Further Limit Theorems 222
8 AN OVERVIEW OF STATISTICAL INFERENCE 227
8.1 The Basics of Point Estimation 228
8.2 The Basics of Interval Estimation 230
8.3 The Basics of Testing Hypotheses 231
8.4 The Basics of Regression Analysis 235
8.5 The Basics of Analysis of Variance 236
8.6 The Basics of Nonparametric Inference 238
9 POINT ESTIMATION 240
9.1 Maximum Likelihood Estimation: Motivation
and Examples 240
9.2 Some Properties of Maximum Likelihood Estimates 253
9.3 Uniformly Minimum Variance Unbiased Estimates 261
9.4 Decision-Theoretic Approach to Estimation 270
9.5 Other Methods of Estimation 277
10 CONFIDENCE INTERVALS AND CONFIDENCE
REGIONS 281
10.1 Conﬁdence Intervals 282
10.2 Conﬁdence Intervals in the Presence of Nuisance
Parameters 289

Contents ix
10.3 A Confidence Region for (μ, σ2
) in the N(μ, σ2
)
Distribution 292
10.4 Confidence Intervals with Approximate Confidence
Coefficient 294
11 TESTING HYPOTHESES 299
11.1 General Concepts, Formulation of Some Testing
Hypotheses 300
11.2 Neyman–Pearson Fundamental Lemma, Exponential Type
Families, Uniformly Most Powerful Tests for Some
Composite Hypotheses 302
11.3 Some Applications of Theorems 2 and 3 315
11.4 Likelihood Ratio Tests 324
12 MORE ABOUT TESTING HYPOTHESES 343
12.1 Likelihood Ratio Tests in the Multinomial Case
and Contingency Tables 343
12.2 A Goodness-of-Fit Test 349
12.3 Decision-Theoretic Approach to Testing Hypotheses 353
12.4 Relationship Between Testing Hypotheses and
Confidence Regions 360
13 A SIMPLE LINEAR REGRESSION MODEL 363
13.1 Setting-up the Model — The Principle of Least Squares 364
13.2 The Least Squares Estimates of β1 and β2, and Some
of Their Properties 366
13.3 Normally Distributed Errors: MLE’s of β1, β2, and σ2
,
Some Distributional Results 374
13.4 Confidence Intervals and Hypotheses Testing Problems 383
13.5 Some Prediction Problems 389
13.6 Proof of Theorem 5 393
13.7 Concluding Remarks 395
14 TWO MODELS OF ANALYSIS OF VARIANCE 397
14.1 One-Way Layout with the Same Number of Observations
per Cell 398
14.2 A Multicomparison Method 407
14.3 Two-Way Layout with One Observation per Cell 412
15 SOME TOPICS IN NONPARAMETRIC INFERENCE 428
15.1 Some Confidence Intervals with Given Approximate
Confidence Coefficient 429
15.2 Confidence Intervals for Quantiles of a Distribution
Function 431

x Contents
15.3 The Two-Sample Sign Test 433
15.4 The Rank Sum and the Wilcoxon–Mann–Whitney
Two-Sample Tests 435
15.5 Nonparametric Curve Estimation 442
APPENDIX 450
SOME NOTATION AND ABBREVIATIONS 480
ANSWERS TO EVEN-NUMBERED EXERCISES 483
INDEX 515

Preface
Overview
This book is an introductory textbook in probability and statistical inference.
No prior knowledge of either probability or statistics is required, although
prior exposure to an elementary precalculus course would prove beneficial in
the sense that the student would not see the basic concepts discussed here for
the first time.
The mathematical prerequisite is a year of calculus and familiarity with
the basic concepts and some results of linear algebra. Elementary differential
and integral calculus will suffice for the majority of the book. In some parts,
such as Chapters 4, 5, and 6, the concept of a multiple integral is used. Also,
in Chapter 6, the student is expected to be at least vaguely familiar with the
basic techniques of changing variables in a single or a multiple integral.
Chapter Descriptions
The material discussed in this book is enough for a one-year course in introduc-
tory probability and statistical inference. It consists of a total of 15 chapters.
Chapters 1 through 7 are devoted to probability, distributional theory, and
related topics. Chapters 9 through 14 discuss the standard topics of para-
metric statistical inference, namely point estimation, interval estimation, and
testing hypotheses. This is done first in a general setting and then in the special
models of linear regression and analysis of variance. Chapter 15 is devoted to
discussing selected topics from nonparametric inference.
Features
This book has a number of features that differentiate it from existing books.
First, the material is arranged in such a manner that Chapters 1 through 8 can
be used independently for an introductory course in probability. The desirable
duration for such a course would be a semester, although a quarter would
xi

xii Preface
also be long enough if some of the proofs were omitted. Chapters 1 though 7
would suffice for this purpose. The centrally placed Chapter 8 plays a twofold
role. First, it serves as a window into what statistical inference is all about
for those taking only the probability part of the course. Second, it paints a
fairly broad picture of the material discussed in considerable detail in the
subsequent chapters. Accordingly and purposely, no specific results are stated,
no examples are discussed, no exercises are included. All these things are done
in the chapters following it. As already mentioned, the sole objective here is
to take the reader through a brief orientation trip to statistical inference; to
indicate why statistical inference is needed in the first place, how the relevant
main problems are formulated, and how we go about resolving them.
The second differentiating feature of the book is the relatively large
number of examples discussed in detail. There are more than 220 such exam-
ples, not including scores of numerical examples and applications. The first
chapter alone is replete with 44 examples selected from a variety of applica-
tions. Their purpose is to impress upon the student the breadth of applications
of probability and statistics, to draw attention to the wide range of applica-
tions where probabilistic and statistical questions are pertinent. At this stage,
one could not possibly provide answers to the questions posed without the
methodology developed in the subsequent chapters. Answers to these ques-
tions are given in the form of examples and exercises throughout the remaining
chapters.
The book contains more than 560 exercises placed strategically at the ends
of sections. The exercises are closely related to the material discussed in the
respective sections, and they vary in the degree of difficulty. Detailed solutions
toallofthemareavailableintheformofa Solutions Manualfortheinstructors
of the course, when this textbook is used. Brief answers to even-numbered
exercises are provided at the end of the book. Also included in the textbook
are approximately 60 figures that help illustrate some concepts and operations.
Still another desirable feature of this textbook is the effort made to mini-
mize the so-called arm waving. This is done by providing a substantial number
of proofs, without ever exceeding the mathematical prerequisites set. This
also helps ameliorate the not so unusual phenomenon of insulting students’
intelligence by holding them incapable of following basic reasoning.
Regardless of the effort made by the author of an introductory book in
probability and statistics to cover the largest possible number of areas where
probability and statistics apply, such a goal is unlikely to be attained. Conse-
quently, no such textbook will ever satisfy students who focus exclusively on
their own area of interest. It is also expected that this book will come as a
disappointment to students who are oriented more toward vocational training
rather than college or university education. This book is not meant to codify
answers to questions in the form of framed formulas and prescription recipes.
Rather, its purpose is to introduce the student to a thinking process and guide
her or him toward the answer sought to a posed question. To paraphrase a
Chinese saying, if you are taught how to fish, you eat all the time, whereas if
you are given a fish, you eat only once.

Preface xiii
Onseveraloccasionsthereaderisreferredforproofsandmorecomprehen-
sive treatment of some topics to the book A Course in Mathematical Statis-
tics, 2nd
edition (1997), Academic Press, by G.G. Roussas. This reference book
was originally written for the same audience as that of the present book. How-
ever, circumstances dictated the adjustment of the level of the reference book
to match the mathematical preparation of the anticipated audience.
On the practical side, a number of points of information are given here.
Thus, logx (logarithm of x), whenever it occurs, is always the natural logarithm
of x (the logarithm of x with base e), whether it is explicitly stated or not.
The rule followed in the use of decimal numbers is that we retain three
decimal digits, the last of which is rounded up to the next higher number, if
the fourth omitted decimal is greater or equal 5. An exemption to this rule is
made when the division is exact, and also when the numbers are read out of
tables. The book is supplied with an appendix consisting of excerpts of tables:
Binomial tables, Poisson tables, Normal tables, t-tables, Chi-Square tables,
and F-tables. The last table, Table 7, consists of a list of certain often-occurring
distributionsalongwithsomeoftheircharacteristics.Theappendixisfollowed
by a list of some notation and abbreviations extensively used throughout the
book, and the body of the book is concluded with brief answers to the even-
numbered exercises.
In closing, a concerted effort has been made to minimize the number of
inevitable misprints and oversights in the book. We have no illusion, however,
that the book is free of them. This author would greatly appreciate being
informed of any errors; such errors will be corrected in a subsequent printing
of the book.
Acknowledgments and Credits
I would like to thank Subhash Bagui, University of West Florida; Matthew
Carlton, Cal Polytechnic State University; Tapas K. Das, University of South
Florida; Jay Devore, Cal Polytechnic State University; Charles Donaghey, Uni-
versity of Houston; Pat Goeters, Auburn University; Xuming He, University of
Illinois, and Krzysztof M. Ostaszewski, Illinois State University, Champaign-
Urbana for their many helpful comments.
Some of the examples discussed in this book have been taken and/or
adapted from material included in thebook Statistics:Principlesand Methods,
2nd edition (1992), [ISBN: 0471548421], by R. A. Johnson, G. K. Bhattacharyya,
Copyright c
1987, 1992, by John Wiley Sons, Inc., and are reprinted by per-
mission of John Wiley Sons, Inc. They are Table 4 on page 74, Examples 8,
1, 2, 4, 12, 4, 2, 1, and 7 on pages 170, 295, 296, 353, 408, 439, 510, 544, and
562, respectively, and Exercises 4.18, 3.19, 4.21, 5.22, 5.34, 8.16, 4.14, 6.34, 3.16,
6.6 and 3.8 on pages 123, 199, 217, 222, 225, 265, 323, 340, 356, 462, and 525,
respectively. The reprinting permission is kindly acknowledged herewith.

Chapter 1
Some Motivating
Examples and Some
Fundamental
Concepts
This chapter consists of three sections. The first section is devoted to present-
ing a number of examples (25 to be precise), drawn from a broad spectrum
of human activities. Their purpose is to demonstrate the wide applicability of
probability and statistics. In the formulation of these examples, certain terms,
such as at random, average, data fit by a line, event, probability (estimated
probability, probability model), rate of success, sample, and sampling (sample
size), are used. These terms are presently to be understood in their everyday
sense, and will be defined precisely later on.
In the second section, some basic terminology and fundamental quantities
are introduced and are illustrated by means of examples. In the closing section,
the concept of a random variable is defined and is clarified through a number
of examples.
1.1 Some Motivating Examples
EXAMPLE 1 In a certain state of the Union, n landfills are classified according to their
concentration of three hazardous chemicals: arsenic, barium, and mercury.
Suppose that the concentration of each one of the three chemicals is charac-
terized as either high or low. Then some of the questions which can be posed
are as follows: (i) If a landfill is chosen at random from among the n, what
is the probability it is of a specific configuration? In particular, what is the
probability that it has: (a) High concentration of barium? (b) High concentra-
tion of mercury and low concentration of both arsenic and barium? (c) High
1

2 Chapter 1 Some Motivating Examples and Some Fundamental Concepts
concentration of any two of the chemicals and low concentration of the third?
(d) High concentration of any one of the chemicals and low concentration of
the other two? (ii) How can one check whether the proportions of the landfills
falling into each one of the eight possible configurations (regarding the levels
of concentration) agree with a priori stipulated numbers?
EXAMPLE 2 Suppose a disease is present in 100p1% (0 p1 1) of a population. A diag-
nostic test is available but is yet to be perfected. The test shows 100p2% false
positives (0 p2 1) and 100p3% false negatives (0 p3 1). That is, for a
patient not having the disease, the test shows positive (+) with probability p2
and negative (−) with probability 1 − p2. For a patient having the disease, the
test shows “−” with probability p3 and “+” with probability 1− p3. A person is
chosen at random from the target population, and let D be the event that the
person is diseased and N be the event that the person is not diseased. Then, it
is clear that some important questions are as follows: In terms of p1, p2, and
p3: (i) Determine the probabilities of the following configurations: D and +,
D and −, N and +, N and −. (ii) Also, determine the probability that a person
will test + or the probability the person will test −. (iii) If the person chosen
tests +, what is the probability that he/she is diseased? What is the probability
that he/she is diseased, if the person tests −?
EXAMPLE 3 In the circuit drawn below, suppose that switch i = 1, . . . , 5 turns on with prob-
ability pi and independently of the remaining switches. What is the probability
of having current transferred from point A to point B?
A B
1 2
5
4 3
EXAMPLE 4 A travel insurance policy pays $1,000 to a customer in case of a loss due to
theft or damage on a 5-day trip. If the risk of such a loss is assessed to be 1 in
200, what is a fair premium for this policy?
EXAMPLE 5 Jones claims to have extrasensory perception (ESP). In order to test the claim,
a psychologist shows Jones five cards that carry different pictures. Then Jones
is blindfolded and the psychologist selects one card and asks Jones to identify

the picture. This process is repeated n times. Suppose, in reality, that Jones
has no ESP but responds by sheer guesses.
(i) Decide on a suitable probability model describing the number of correct
responses. (ii) What is the probability that at most n/5 responses are correct?
(iii) What is the probability that at least n/2 responses are correct?
EXAMPLE 6 A government agency wishes to assess the prevailing rate of unemployment
in a particular county. It is felt that this assessment can be done quickly and
effectively by sampling a small fraction n, say, of the labor force in the county.
The obvious questions to be considered here are: (i) What is a suitable prob-
ability model describing the number of unemployed? (ii) What is an estimate
of the rate of unemployment?
EXAMPLE 7 Suppose that, for a particular cancer, chemotherapy provides a 5-year survival
rate of 80% if the disease could be detected at an early stage. Suppose further
that npatients, diagnosed to have this form of cancer at an early stage, are just
starting the chemotherapy. Finally, let X be the number of patients among the
n who survive 5 years.
Then the following are some of the relevant questions which can be asked:
(i) What are the possible values of X, and what are the probabilities that each
one of these values is taken on? (ii) What is the probability that X takes values
between two speciﬁed numbers a and b, say? (iii) What is the average number
of patients to survive 5 years, and what is the variation around this average?
EXAMPLE 8 An advertisement manager for a radio station claims that over 100p% (0 p
1)of all young adults in the city listen to a weekend music program. To establish
this conjecture, a random sample of size n is taken from among the target
population and those who listen to the weekend music program are counted.
(i) Decide on a suitable probability model describing the number of young
adults who listen to the weekend music program. (ii) On the basis of the
collected data, check whether the claim made is supported or not. (iii) How
large a sample size nshould be taken to ensure that the estimated average and
the true proportion do not differ in absolute value by more than a speciﬁed
number with prescribed (high) probability?
EXAMPLE 9 When the output of a production process is stable at an acceptable standard,
it is said to be “in control.” Suppose that a production process has been in
control for some time and that the proportion of defectives has been p. As
a means of monitoring the process, the production staff will sample n items.
Occurrence of k or more defectives will be considered strong evidence for “out
of control.”
(i) Decide on a suitable probability model describing the number X of defec-
tives; what are the possible values of X, and what is the probability that each of

these values is taken on? (ii) On the basis of the data collected, check whether
or not the process is out of control. (iii) How large a sample size n should be
taken to ensure that the estimated proportion of defectives will not differ in
absolute value from the true proportion of defectives by more than a specified
quantity with prescribed (high) probability?
EXAMPLE 10 An electronic scanner is believed to be more efficient in determining flaws in
a material than a mechanical testing method which detects 100p% (0 p 1)
of the flawed specimens. To determine its success rate, n specimens with
flaws are tested by the electronic scanner.
(i) Decide on a suitable probability model describing the number X of the
flawed specimens correctly detected by the electronic scanner; what are the
possible values of X, and what is the probability that each one of these values
is taken on? (ii) Suppose that the electronic scanner detects correctly k out of
nflawed specimens. Check whether or not the rate of success of the electronic
scanner is higher than that of the mechanical device.
EXAMPLE 11 At a given road intersection, suppose that X is the number of cars passing by
until an observer spots a particular make of a car (e.g., a Mercedes).
Then some of the questions one may ask are as follows: (i) What are the
possible values of X? (ii) What is the probability that each one of these values
is taken on? (iii) How many cars would the observer expect to observe until
the first Mercedes appears?
EXAMPLE 12 A city health department wishes to determine whether the mean bacteria count
per unit volume of water at a lake beach is within the safety level of 200. A
researcher collected nwater samples of unit volume and recorded the bacteria
counts.
Relevant questions here are: (i) What is the appropriate probability model
describing the number X of bacteria in a unit volume of water; what are the
possible values of X, and what is the probability that each one of these values is
taken on? (ii) Do the data collected indicate that there is no cause for concern?
EXAMPLE 13 Consider an aptitude test administered to aircraft pilot trainees, which requires
a series of operations to be performed in quick succession.
Relevant questions here are: (i) What is the appropriate probability model for
the time required to complete the test? (ii) What is the probability that the test
is completed in no less than t1 minutes, say? (iii) What is the percentage of
candidates passing the test, if the test is to be completed within t2 minutes, say?
EXAMPLE 14 Measurements of the acidity (pH) of rain samples were recorded at n sites in
an industrial region.

(i) Decide on a suitable probability model describing the number X of the
acidity of rain measured. (ii) On the basis of the measurements taken, provide
an estimate of the average acidity of rain in that region.
EXAMPLE 15 To study the growth of pine trees at an early state, a nursery worker records n
measurements of the heights of 1-year-old red pine seedlings.
(i) Decide on a suitable probability model describing the heights X of the pine
seedlings. (ii) On the basis of the n measurements taken, determine average
height of the pine seedlings. (iii) Also, check whether these measurements
support the stipulation that the average height is a specified number.
EXAMPLE 16 It is claimed that a new treatment is more effective than the standard treatment
for prolonging the lives of terminal cancer patients. The standard treatment
has been in use for a long time, and from records in medical journals the mean
survival period is known to have a certain numerical value (in years). The
new treatment is administered to n patients, and their duration of survival is
recorded.
(i) Decide on suitable probability models describing the survival times X and
Y under the old and the new treatments, respectively. (ii) On the basis of the
existing journal information and the data gathered, check whether or not the
claim made is supported.
EXAMPLE 17 A medical researcher wishes to determine whether a pill has the undesirable
side effect of reducing the blood pressure of the user. The study requires
recording the initial blood pressures of n college-age women. After the use of
the pill regularly for 6 months, their blood pressures are again recorded.
(i) Decide on suitable probability models describing the blood pressures, ini-
tially and after the 6-month period. (ii) Do the observed data support the claim
that the use of the pill reduces blood pressure?
EXAMPLE 18 It is known that human blood is classified in four types denoted by A, B, AB,
and O. Suppose that the blood of n persons who have volunteered to donate
blood at a plasma center has been classified in these four categories. Then a
number of questions can be posed; some of them are:
(i) What is the appropriate probability model to describe the distribution of
the blood types of the n persons into the four types? (ii) What is the esti-
mated probability that a person, chosen at random from among the n, has
a specified blood type (e.g., O)? (iii) What are the proportions of the n per-
sons falling into each one of the four categories? (iv) How can one check
whether the observed proportions are in agreement with a priori stipulated
numbers?

EXAMPLE 19 The following record shows a classiﬁcation of 41,208 births in Wisconsin
(courtesy of Professor Jerome Klotz). Set up a suitable probability model and
check whether or not the births are uniformly distributed over all 12 months
of the year.
Jan. 3,478 July 3,476
Feb. 3,333 Aug. 3,495
March 3,771 Sept. 3,490
April 3,542 Oct. 3,331
May 3,479 Nov. 3,188
June 3,304 Dec. 3,321
Total 41,208
EXAMPLE 20 To compare the effectiveness of two diets A and B, 150 infants were included
in a study. Diet A was given to 80 randomly selected infants and diet B was
given to the other 70 infants. At a later time, the health of each infant was
observed and classiﬁed into one of the three categories: “excellent,” “average,”
and “poor.” The frequency counts are tabulated as follows:
HEALTH UNDER TWO DIFFERENT DIETS
Excellent Average Poor Sample Size
Diet A 37 24 19 80
Diet B 17 33 20 70
Total 54 57 39 150
Set up a suitable probability model for this situation, and, on the basis of the
observed data, compare the effectiveness of the two diets.
EXAMPLE 21 Osteoporosis (loss of bone minerals) is a common cause of broken bones in
the elderly. A researcher on aging conjectures that bone mineral loss can be
reduced by regular physical therapy or by certain kinds of physical activity. A
study is conducted on nelderly subjects of approximately the same age divided
into control, physical therapy, and physical activity groups. After a suitable
period of time, the nature of change in bone mineral content is observed.
Set up a suitable probability model for the situation under consideration, and
check whether or not the observed data indicate that the change in bone
mineral varies for different groups.
CHANGE IN BONE MINERAL
Appreciable Little Appreciable
Loss Change Increase Total
Control 38 15 7 60
Therapy 22 32 16 70
Activity 15 30 25 70
Total 75 77 48 200

EXAMPLE 22 In the following table, the data x = undergraduate GPA and y = score in the
Graduate Management Aptitude Test (GMAT) are recorded.
DATA OF UNDERGRADUATE GPA (x)
AND GMAT SCORE (y)
x y x y x y
3.63 447 2.36 399 2.80 444
3.59 588 2.36 482 3.13 416
3.30 563 2.66 420 3.01 471
3.40 553 2.68 414 2.79 490
3.50 572 2.48 533 2.89 431
3.78 591 2.46 509 2.91 446
3.44 692 2.63 504 2.75 546
3.48 528 2.44 336 2.73 467
3.47 552 2.13 408 3.12 463
3.35 520 2.41 469 3.08 440
3.39 543 2.55 538 3.03 419
3.00 509
(i) Draw a scatter plot of the pairs (x, y). (ii) On the basis of part (i), set up
a reasonable model for the representation of the pairs (x, y). (iii) Indicate
roughly how this model can be used to predict a GMAT score on the basis of
the corresponding GPA score.
EXAMPLE 23 In an experiment designed to determine the relationship between the doses
of a compost fertilizer x and the yield y of a crop, n values of x and y are
observed. On the basis of prior experience, it is reasonable to assume that the
pairs (x, y) are fitted by a straight line, which can be determined by certain
summary values of the data. Later on, it will be seen how this is specifically
done and also how this model can be used for various purposes, including that
of predicting a value of y on the basis of a given value of x.
EXAMPLE 24 In an effort to improve the quality of recording tapes, the effects of four kinds
of coatings A, B, C, and D on the reproducing quality of sound are compared.
Twentytwomeasurementsofsounddistortionsaregiveninthefollowingtable.
SOUND DISTORTIONS OBTAINED
WITH FOUR TYPES OF COATINGS
Coating Observations
A 10, 15, 8, 12, 15
B 14, 18, 21, 15
C 17, 16, 14, 15, 17, 15, 18
D 12, 15, 17, 15, 16, 15
In connection with these data, several questions may be posed (and will be
posed later on). The most immediate of them all is the question of whether
or not the data support the existence of any significant difference among the
average distortions obtained using the four coatings.

EXAMPLE 25 Charles Darwin performed an experiment to determine whether self-fertilized
and cross-fertilized plants have different growth rates. Pairs of Zea mays
plants, one self- and the other cross-fertilized, were planted in pots, and their
heights were measured after a speciﬁed period of time. The data Darwin ob-
tained were:
PLANT HEIGHT (IN 1/8 INCHES)
Pair Cross- Self- Pair Cross- Self-
1 188 139 9 146 132
2 96 163 10 173 144
3 168 160 11 186 130
4 176 160 12 168 144
5 153 147 13 177 102
6 172 149 14 184 124
7 177 149 15 96 144
8 163 122
Source: Darwin, C., “The Effects of Cross- and Self-Fertilization
in the Vegetable Kingdom,” D. Appleton and Co., New York, 1902.
These data lead to many questions, the most immediate being whether cross-
fertilized plants have a higher growth rate than self-fertilized plants. This ex-
ample will be revisited later on.
1.2 Some Fundamental Concepts
One of the most basic concepts in probability and statistics is that of a random
experiment. Although a more precise deﬁnition is possible, we will restrict
ourselves here to understanding a random experiment as a procedure which
is carried out under a certain set of conditions; it can be repeated any number
of times under the same set of conditions, and upon the completion of the
procedure certain results are observed. The results obtained are denoted by s
and are called sample points. The set of all possible sample points is denoted
by S and is called a sample space. Subsets of S are called events and are
denoted by capital letters A, B, C, etc. An event consisting of one sample point
only, {s}, is called a simple event and composite otherwise. An event A occurs
(or happens) if the outcome of the random experiment (that is, the sample
point s) belongs in A, s ∈ A; A does not occur (or does not happen) if s /
∈ A.
The event S always occurs and is called the sure or certain event. On the
other hand, the event Ø never happens and is called the impossible event. Of
course, the relation A ⊆ B between two events A and B means that the event
B occurs whenever A does, but not necessarily the opposite. (See Figure 1.1
for the Venn diagram depicting the relation A ⊆ B.) The events A and B are
equal if both A ⊆ B and B ⊆ A.
Some random experiments are given in the following along with corre-
sponding sample spaces and some events.

A
s1 •
B
• s2
S
Figure 1.1
A ⊆ B; in Fact,
A ⊂ B, Because
s2 ∈ B, But s2 ∈ A
EXAMPLE 26 Tossing three distinct coins once.
Then, with H and T standing for “heads” and “tails,” respectively, a sample
space is:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
The event A = “no more than 1 H occurs” is given by:
A = {TTT, HTT, THT, TTH}.
EXAMPLE 27 Rolling once two distinct dice.
Then a sample space is:
S = {(1, 1), (1, 2), . . . , (1, 6), . . . , (6, 1), (6, 2), . . . , (6, 6)},
and the event B = “the sum of numbers on the upper faces is ≤ 5” is:
B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}.
EXAMPLE 28 Drawing a card from a well-shufﬂed standard deck of 52 cards. Denoting by
C, D, H, and S clubs, diamonds, hearts, and spades, respectively, by J, Q, K
Jack, Queen, and King, and using 1 for aces, the sample space is given by:
S = {1C , . . . , 1S, . . . , 10C , . . . , 10S, . . . , KC , . . . , KS}.
An event A may be described by: A = “red and face card,” so that
A = {JD, JH, QD, QH, KD, KH}.
EXAMPLE 29 Drawing (without replacement) two balls from an urn containing mnumbered
black balls and n numbered red balls.

Then, in obvious notation, a sample space here is:
S = {b1b2, . . . , b1bm, . . . , bmb1, . . . , bmbm−1,
b1r1, . . . , b1rn, . . . , bmr1, . . . , bmrn,
r1b1, . . . , r1bm, . . . , rnb1, . . . , rnbm,
r1r2, . . . , r1rn, . . . , rnr1, . . . , rnrn−1}.
An event A may be the following: A = “the sum of the numbers on the balls
does not exceed 4.” Then
A = {b1b2, b1b3, b2b1, b3b1, b1r1, b1r2, b1r3,
b2r1, b2r2, b3r1, r1b1, r1b2, r1b3, r2b1,
r2b2, r3b1, r1r2, r1r3, r2r1, r3r1} (assuming that m, n ≥ 3).
EXAMPLE 30 Recording the gender of children of two-children families.
With b and g standing for boy and girl, and with the first letter on the left
denoting the older child, a sample space is: S = {bb, bg, gb, gg}. An event B
may be: B = “children of both genders.” Then B = {bg, gb}.
EXAMPLE 31 Ranking five horses in a horse race.
Then the suitable sample space S consists of 120 sample points, corresponding
to the 120 permutations of the numbers 1, 2, 3, 4, 5. (We exclude ties.) The event
A = “horse #3 comes second” consists of the 24 sample points, where 3 always
occurs in the second place.
EXAMPLE 32 Tossing a coin repeatedly until H appears for the first time.
The suitable sample space here is:
S = {H, TH, TTH, . . . , TT . . . TH, . . .}.
Then the event A = “the 1st H does not occur before the 10th tossing” is given
by:
A =

T . . . T

9
H, T . . . T

10
H, . . .

.
EXAMPLE 33 Recording the number of telephone calls served by a certain telephone ex-
change center within a specified period of time.
Clearly, the sample space here is: S = {0, 1, . . . , C}, where C is a suitably
large number associated with the capacity of the center. For mathematical
convenience, we often take S to consist of all nonnegative integers; that is,
S = {0, 1, . . .}.

EXAMPLE 34 Recording the number of traffic accidents which occurred in a specified loca-
tion within a certain period of time.
As in the previous example, S = {0, 1, . . . , M} for a suitable number M. If M
is sufficiently large, then S is taken to be: S = {0, 1, . . .}.
EXAMPLE 35 Recording the number of particles emitted by a certain radioactive source
within a specified period of time.
As in the previous two examples, S is taken to be: S = {0, 1, . . . , M}, where M
is often a large number, and then as before S is modified to be: S = {0, 1, . . .}.
EXAMPLE 36 Recording the lifetime of an electronic device, or of an electrical appliance,
etc.
Here S is the interval (0, T) for some reasonable value of T; that is, S = (0, T).
Sometimes, for justifiable reasons, we take, S = (0, ∞).
EXAMPLE 37 Recording the distance from the bull’s eye of the point where a dart, aiming at
the bull’s eye, actually hits the plane. Here it is clear that S = (0, ∞).
EXAMPLE 38 Measuring the dosage of a certain medication, administered to a patient, until
a positive reaction is observed.
Here S = (0, D) for some suitable D (not rendering the medication lethal!).
EXAMPLE 39 Recording the yearly income of a target population.
If the incomes are measured in $ and cents, the outcomes are fractional num-
bers in an interval [0, M] for some reasonable M. Again, for reasons similar to
those cited in Example 36, S is often taken to be S = [0, ∞).
EXAMPLE 40 Waiting until the time the Dow–Jones Industrial Average index reaches or
surpasses a specified level.
Here, with reasonable qualifications, we may chose to take S = (0, ∞).
Examples 1–25, suitably interpreted, may also serve as further illustrations of
random experiments. All examples described previously will be revisited on
various occasions.
For instance, in Example 1 and in self-explanatory notation, a suitable sample
space is:
S = {Ah Bh Mh, Ah Bh M, Ah B Mh, A Bh Mh, Ah B M,
A Bh M, A B Mh, A B M}.

Then the events A = “no chemical occurs at high level” and B = “at least two
chemicals occur at high levels” are given by:
A = {A B M}, B = {Ah Bh M, Ah B Mh, A Bh Mh, Ah Bh Mh}.
In Example 2, a patient is classified according to the result of the test, giving
rise to the following sample space:
S = {D+, D−, N+, N−},
where D and N stand for the events “patient has the disease” and “patient
does not have the disease,” respectively. Then the event A = “false diagnosis
of test” is given by: A = {D−, N+}.
In Example 5, the suitable probability model is the so-called binomial model.
The sample space S is the set of 2n
points, each point consisting of a sequence
of n S’s and F’s, S standing for success (on behalf of Jones) and F standing for
failure. Then the questions posed can be answered easily.
Examples 6 through 10 can be discussed in the same framework as that of
Example 5 with obvious modifications in notation.
In Example 11, a suitable sample space is:
S = {M, Mc
M, Mc
Mc
M, . . . , Mc
· · · Mc
M, . . .},
where M stands for the passing by of a Mercedes car. Then the events A and
B, where A = “Mercedes was the 5th car passed by” and B = “Mercedes was
spotted after the first 3 cars passed by” are given by:
A = {Mc
Mc
Mc
Mc
M} and B = {Mc
Mc
Mc
M, Mc
Mc
Mc
Mc
M, . . .}.
In Example 12, a suitable sample space is: S = {0, 1, . . . , M} for an appropri-
ately large (integer) M; for mathematical convenience, S is often taken to be:
S = {0, 1, 2, . . .}.
In Example 13, a suitable sample space is: S = (0, T) for some reasonable
value of T. In such cases, if T is very large, mathematical convenience dictates
replacement of the previous sample space by: S = (0, ∞).
Examples 14 and 15 can be treated in the same framework as Example 13 with
obvious modifications in notation.
In Example 18, a suitable sample space S is the set of 4n
points, each point
consisting of a sequence of n symbols A, B, AB, and O. The underlying prob-
ability model is the so-called multinomial model, and the questions posed can
be discussed by available methodology. Actually, there is no need even to refer
to the sample space S. All one has to do is to consider the outcomes in the n
trials and then classify the n outcomes into four categories A, B, AB, and O.

Example 19 fits into the same framework as that of Example 18. Here the
suitable S consists of 1241,208
points, each point being a sequence of symbols
representing the 12 months. As in the previous example, there is no need,
however,eventorefertothissamplespace.Example20isalsoofthesametype.
In many cases, questions posed can be discussed without reference to any
explicit sample space. This is the case, for instance, in Examples 16–17 and
21–25.
In the examples discussed previously, we have seen sample spaces consisting
of finitely many sample points (Examples 26–31), sample spaces consisting of
countably infinite many points (for example, as many as the positive integers)
(Example 32 and also Examples 33–35 if we replace C and M by ∞ for mathe-
matical convenience), and sample spaces consisting of as many sample points
as there are in a nondegenerate finite or infinite interval in the real line, which
interval may also be the entire real line (Examples 36–40). Sample spaces with
countably many points (i.e., either finitely many or countably infinite many)
are referred to as discrete sample spaces. Sample spaces with sample points
as many as the numbers in a nondegenerate finite or infinite interval in the real
line = (−∞, ∞) are referred to as continuous sample spaces.
Returning now to events, when one is dealing with them, one may perform
the same operations as those with sets. Thus, the complement of the event A,
denoted by Ac
, is the event defined by: Ac
= {s ∈ S; s /
∈ A}. The event Ac
is
presented by the Venn diagram in Figure 1.2. So Ac
occurs whenever A does
not, and vice versa.
S
A
Ac
Figure 1.2
Ac
Is the Shaded
Region
The union of the events A1, . . . , An, denoted by A1 ∪. . .∪ An or
n
j=1 Aj, is the
event defined by
n
j=1 Aj = {s ∈ S; s ∈ Aj, for at least one j = 1, . . . , n}. So
the event
n
j=1 Aj occurs whenever at least one of Aj, j = 1, . . . , noccurs. For
n = 2, A1 ∪ A2 is presented in Figure 1.3. The definition extends to an infinite
number of events. Thus, for countably infinite many events Aj, j = 1, 2, . . . ,
one has
∞
j=1 Aj = {s ∈ S; s ∈ Aj, for at least one j = 1, 2, . . .}.
The intersection of the events Aj, j = 1, . . . , n is the event denoted by
A1 ∩ · · · ∩ An or n
j=1 Aj and is defined by n
j=1 Aj = {s ∈ S; s ∈ Aj, for
all j = 1, . . . , n}. Thus, n
j=1 Aj occurs whenever all Aj, J = 1, . . . , n

A1 A2
S
Figure 1.3
A1 ∪ A2 Is the
Shaded Region
A1 A2
S
Figure 1.4
A1 ∩ A2 Is the
Shaded Region
occur simultaneously. For n = 2, A1 ∩ A2 is presented in Figure 1.4. This
definition extends to an infinite number of events. Thus, for countably infi-
nite many events Aj, j = 1, 2, . . . , one has ∞
j=1 Aj = {s ∈ S; s ∈ Aj, for all
j = 1, 2, . . .}.
If A1 ∩ A2 = Ø, the events A1 and A2 are called disjoint (see Figure 1.5).
The events Aj, j = 1, 2, . . . , are said to be mutually or pairwise disjoint, if
Ai ∩ Aj = Ø whenever i = j.
A1 A2
S
Figure 1.5
A1 and A2 Are
Disjoint; That Is
Ai ∩ Aj = Ø
The differences A1 − A2 and A2 − A1 are the events defined by A1 − A2 =
{s ∈ S; s ∈ A1, s /
∈ A2}, A2 − A1 = {s ∈ S; s ∈ A2, s /
∈ A1} (see Figure 1.6).
From the definition of the preceding operations, the following properties fol-
low immediately, and they are listed here for reference.
1. Sc
= Ø, Øc
= S, (Ac
)c
= A.
2. S ∪ A = S, Ø ∪ A = A, A ∪ Ac
= S, A ∪ A = A.
3. S ∩ A = A, Ø ∩ A = Ø, A ∩ Ac
= Ø, A ∩ A = A.

A1 A2
S
Figure 1.6
A1 − A2 Is ,
A2 − A1 Is

The previous statements are all obvious, as is the following: Ø ⊆ A for every
event A in S. Also,
4. A1 ∪ (A2 ∪ A3) = (A1 ∪ A2) ∪ A3
A1 ∩ (A2 ∩ A3) = (A1 ∩ A2) ∩ A3
(associative laws)
5. A1 ∪ A2 = A2 ∪ A1
A1 ∩ A2 = A2 ∩ A1
(commutative laws)
6. A ∩ (∪j Aj) = ∪j(A ∩ Aj)
A ∪ (∩j Aj) = ∩j(A ∪ Aj)
(distributive laws)
In the last relations, as well as elsewhere, when the range of the index j is
not indicated explicitly, it is assumed to be a finite set, such as {1, . . . , n}, or a
countably infinite set, such as {1, 2, . . .}.
For the purpose of demonstrating some of the set-theoretic operations just
defined, let us consider some further concrete examples.
EXAMPLE 41 Consider the sample space S = {s1, s2, s3, s4, s5, s6, s7, s8} and define the events
A1, A2, and A3 as follows: A1 = {s1, s2, s3}, A2 = {s2, s3, s4, s5}, A3 = {s3, s4,
s5, s8}. Then observe that:
Ac
1 = {s4, s5, s6, s7, s8}, Ac
2 = {s1, s6, s7, s8}, Ac
3 = {s1, s2, s6, s7};
A1 ∪ A2 = {s1, s2, s3, s4, s5}, A1 ∪ A3 = {s1, s2, s3, s4, s5, s8},
A2 ∪ A3 = {s2, s3, s4, s5, s8}, A1 ∪ A2 ∪ A3 = {s1, s2, s3, s4, s5, s8};
A1 ∩ A2 = {s2, s3}, A1 ∩ A3 = {s3}, A1 ∩ A2 ∩ A3 = {s3};
A1 − A2 = {s1}, A2 − A1 = {s4, s5}, A1 − A3 = {s1, s2},
A3 − A1 = {s4, s5, s8}, A2 − A3 = {s2}, A3 − A2 = {s8};
(Ac
1)c
= {s1, s2, s3}(=A1), Ac
2
c
= {s2, s3, s4, s5}(=A2),
Ac
3
c
= {s3, s4, s5, s8}(=A3).
An identity and DeMorgan’s laws stated subsequently are of significant impor-
tance. Their justifications are left as exercises (see Exercises 2.14 and 2.15).
An identity ∪j Aj = A1 ∪ Ac
1 ∩ A2 ∪ Ac
1 ∩ Ac
2 ∩ A3 ∪ . . .
EXAMPLE 42 From Example 41, we have:
A1 = {s1, s2, s3}, Ac
1 ∩ A2 = {s4, s5}, Ac
1 ∩ Ac
2 ∩ A3 = {s8},

Note that A1, Ac
1 ∩ A2, Ac
1 ∩ Ac
2 ∩ A3 are pairwise disjoint. Now A1 ∪ (Ac
1 ∩ A2)∪
(Ac
1 ∩ Ac
2 ∩ A3) = {s1, s2, s3, s4, s5, s8}, which is equal to A1 ∪ A2 ∪ A3; that is,
A1 ∪ A2 ∪ A3 = A1 ∪ Ac
1 ∩ A2 ∪ Ac
1 ∩ Ac
2 ∩ A3
as the preceding identity states.
The signiﬁcance of the identity is that the events on the right-hand side are
pairwise disjoint, whereas the original events Aj, j ≥ 1, need not be so.
DeMorgan’s laws (∪j Aj)c
= ∩j Ac
j, (∩j Aj)c
= ∪j Ac
j.
EXAMPLE 43 Again from Example 41, one has:
(A1 ∪ A2)c
= {s6, s7, s8}, Ac
1 ∩ Ac
2 = {s6, s7, s8};
(A1 ∪ A2 ∪ A3)c
= {s6, s7}, Ac
1 ∩ Ac
2 ∩ Ac
3 = {s6, s7};
(A1 ∩ A2)c
= {s1, s4, s5, s6, s7, s8}, Ac
1 ∪ Ac
2 = {s1, s4, s5, s6, s7, s8};
(A1 ∩ A2 ∩ A3)c
= {s1, s2, s4, s5, s6, s7, s8},
Ac
1 ∪ Ac
2 ∪ Ac
3 = {s1, s2, s4, s5, s6, s7, s8},
so that
(A1 ∪ A2)c
= Ac
1 ∩ Ac
2, (A1 ∪ A2 ∪ A3)c
= Ac
1 ∩ Ac
2 ∩ Ac
3, as DeMorgan’s
(A1 ∩ A2)c
= Ac
1 ∪ Ac
2, (A1 ∩ A2 ∩ A3)c
= Ac
1 ∪ Ac
2 ∪ Ac
3, laws state.
As a further demonstration of how complements, unions, and intersections of
sets are used for the expression of new sets, consider the following example.
EXAMPLE 44 In terms of the events A1, A2, and A3 (in some sample space S) and, perhaps,
their complements, unions, and intersections, express the following events:
Di = “Ai does not occur,” i = 1, 2, 3, so that D1 = Ac
1, D2 = Ac
2, D3 = Ac
3;
E = “all A1, A2, A3 occur,” so that E = A1 ∩ A2 ∩ A3;
F = “none of A1, A2, A3 occurs,” so that F = Ac
1 ∩ Ac
2 ∩ Ac
3;
G = “at least one of A1, A2, A3 occurs,” so that G = A1 ∪ A2 ∪ A3;
H = “exactly two of A1, A2, A3 occur,” so that H = A1 ∩ A2 ∩ Ac
3 ∪
A1 ∩ Ac
2 ∩ A3 ∪ Ac
1 ∩ A2 ∩ A3 ;
I = “exactly one of A1, A2, A3 occurs,” so that I = A1 ∩ Ac
2 ∩ Ac
3 ∪
Ac
1 ∩ A2 ∩ Ac
3 ∪ Ac
1 ∩ Ac
2 ∩ A3 .
It also follows that:
G = “exactly one of A1, A2, A3 occurs” ∪ “exactly two of A1, A2, A3 occur” ∪
“all A1, A2, A3 occur”
= I ∪ H ∪ E.
This section is concluded with the concept of a monotone sequence of events.
Namely, the sequence of events {An}, n ≥ 1, is said to be monotone, if either

Exercises 17
A1 ⊆ A2 ⊆ . . . (increasing) or A1 ⊇ A2 ⊇ . . . (decreasing). In case of an
increasing sequence, the union
∞
j=1 Aj is called the limit of the sequence, and
in case of a decreasing sequence, the intersection ∞
j=1 Aj is called its limit.
The concept of the limit is also defined, under certain conditions, for non-
monotone sequences of events, but we are not going to enter into it here. The
interested reader is referred to Definition 1, page 5, of the book A Course in
Mathematical Statistics,2ndedition(1997),AcademicPress,byG.G.Roussas.
Exercises
2.1 An airport limousine departs from a certain airport with three passengers
to be delivered in any one of three hotels denoted by H1, H2, H3. Let
(x1, x2, x3) denote the number of passengers left at hotels H1, H2, and
H3, respectively.
(i) Write out the sample space S of all possible deliveries.
(ii) Consider the events A, B, C, and D, defined as follows, and express
them in terms of sample points.
A = “one passenger in each hotel,”
B = “all passengers in H1,”
C = “all passengers in one hotel,”
D = “at least two passengers in H1,”
E = “fewer passengers in H1 than in any one of H2 or H3.”
2.2 A machine dispenses balls which are either red or black or green. Suppose
we operate the machine three successive times and record the color of
the balls dispensed, to be denoted by r, b, and g for the respective colors.
(i) Write out an appropriate sample space S for this experiment.
(ii) Consider the events A, B, and C, defined as follows, and express
them by means of sample points.
A = “all three colors appear,”
B = “only two colors appear,”
C = “at least two colors appear.”
2.3 A university library has five copies of a textbook to be used in a certain
class. Of these copies, numbers 1 through 3 are of the 1st edition, and
numbers 4 and 5 are of the 2nd edition. Two of these copies are chosen
at random to be placed on a 2-hour reserve.
(i) Write out an appropriate sample space S.
A = “both books are of the 1st edition,”
B = “both books are of the 2nd edition,”
C = “one book of each edition,”
D = “no book is of the 2nd edition.”

2.4 A large automobile dealership sells three brands of American cars, de-
noted by a1, a2, a3; two brands of Asian cars, denoted by b1, b2; and one
brand of a European car, denoted by c. We observe the cars sold in two
consecutive sales. Then:
(i) Write out an appropriate sample space for this experiment.
(ii) Express the events defined as follows in terms of sample points:
A = “American brands in both sales,”
B = “American brand in the first sale and Asian brand in the second
sale,”
C = “American brand in one sale and Asian brand in the other sale,”
D = “European brand in one sale and Asian brand in the other sale.”
2.5 Of two gas stations I and II located at a certain intersection, I has five gas
pumps and II has six gas pumps. On a given time of a day, observe the
numbers x and y of pumps in use in stations I and II, respectively.
(i) Write out the sample space S for this experiment.
A = “three pumps are in use in station I,”
B = “the number of pumps in use in both stations is the same,”
C = “the number of pumps in use in station II is larger than that in
station I,”
D = “the total number of pumps in use in both stations is not greater
than 4.”
2.6 At a certain busy airport, denote by A, B, C, and D the events defined as
follows:
A = “at least 5 planes are waiting to land,”
B = “at most 3 planes are waiting to land,”
C = “at most 2 planes are waiting to land,”
D = “exactly 2 planes are waiting to land.”
In terms of the events A, B, C, and D and, perhaps, their complements,
express the following events:
E = “at most 4 planes are waiting to land,”
F = “at most 1 plane is waiting to land,”
G = “exactly 3 planes are waiting to land,”
H = “exactly 4 planes are waiting to land,”
I = “at least 4 planes are waiting to land.”
2.7 Let S = {(x, y) ∈ 2
; − 3 ≤ x ≤ 3, 0 ≤ y ≤ 4, x and y integers}, and
define the events A, B, C, and D as follows:
A = {(x, y) ∈ S; x = y}, B = {(x, y) ∈ S; x = −y},
C = {(x, y) ∈ S; x2
= y2
}, D = {(x, y) ∈ S; x2
+ y2
≤ 5}.
List the members of the events just defined.

1.3 Random Variables 19
2.8 In terms of the events A1, A2, A3 in a sample space S and, perhaps, their
complements, express the following events:
(i) B0 = {s ∈ S; s belongs to none of A1, A2, A3},
(ii) B1 = {s ∈ S; s belongs to exactly one of A1, A2, A3},
(iii) B2 = {s ∈ S; s belongs to exactly two of A1, A2, A3},
(iv) B3 = {s ∈ S; s belongs to all of A1, A2, A3},
(v) C = {s ∈ S; s belongs to at most two of A1, A2, A3},
(vi) D = {s ∈ S; s belongs to at least one of A1, A2, A3}.
2.9 If for three events A, B, and C it happens that either A ∪ B ∪ C = A or
A ∩ B ∩ C = A, what conclusions can you draw?
2.10 Show that A is the impossible event (that is, A = Ø), if and only if
(A ∩ Bc
) ∪ (Ac
∩ B) = B for every event B.
2.11 Let A, B, and C be arbitrary events in S. Determine whether each of the
following statements is correct or incorrect.
(i) (A − B) ∪ B = (A ∩ Bc
) ∪ B = B,
(ii) (A ∪ B) − A = (A ∪ B) ∩ Ac
= B,
(iii) (A ∩ B) ∩ (A − B) = (A ∩ B) ∩ (A ∩ Bc
) = Ø,
(iv) (A ∪ B) ∩ (B ∪ C) ∩ (C ∪ A) = (A ∩ B) ∪ (B ∩ C) ∪ (C ∩ A).
2.12 For any three events A, B, and C in a sample space S show that the
transitive property, A ⊆ B and B ⊆ C, implies that A ⊆ C holds.
2.13 Establish the distributive laws, namely A ∩ (∪j Aj) = ∪j(A ∩ Aj) and
A ∪ (∩j Aj) = ∩j(A ∪ Aj).
2.14 Establish the identity:
∪j Aj = A1 ∪ Ac
1 ∩ A2 ∪ Ac
1 ∩ Ac
2 ∩ A3 ∪ · · ·
2.15 Establish DeMorgan’s laws, namely
(∪j Aj)c
= ∩j Ac
j and (∩j Aj)c
= ∪j Ac
j.
2.16 Let S = and, for n = 1, 2, . . . , deﬁne the events An and Bn by:
An = x ∈ ; − 5 +
1
n
x 20 −
1
n
, Bn x ∈ ; 0 x 7 +
3
n
.
(i) Show that the sequence {An} is increasing and the sequence {Bn} is
decreasing.
(ii) Identify the limits, lim
n→∞
An =
∞
n=1 An and lim
n→∞
Bn = ∞
n=1 Bn.
1.3 Random Variables
For every random experiment, there is at least one sample space appropri-
ate for the random experiment under consideration. In many cases, however,
muchoftheworkcanbedonewithoutreferencetoanexplicitsamplespace.In-
stead, what are used extensively are random variables and their distributions.

Those quantities will be studied extensively in subsequent chapters. What is
done in this section is the introduction of the concept of a random variable.
Formally, a random variable, to be shortened to r.v., is simply a function
defined on a sample space S and taking values in the real line = (−∞, ∞).
Random variables are denoted by capital letters, such as X, Y, Z, with or with-
out subscripts. Thus, the value of the r.v. X at the sample point s is X(s), and
the set of all values of X, that is, the range of X, is usually denoted by X(S).
The only difference between a r.v. and a function in the usual calculus sense
is that the domain of a r.v. is a sample space S, which may be an abstract set,
unlike the usual concept of a function, whose domain is a subset of or of a
Euclidean space of higher dimension. The usage of the term “random variable”
employed here rather than that of a function may be explained by the fact that
a r.v. is associated with the outcomes of a random experiment. Thus, one may
argue that X(s) is not known until the random experiment is actually carried
out and s becomes available. Of course, on the same sample space, one may
define many distinct r.v.’s.
In reference to Example 26, instead of the sample space S exhibited there,
one may be interested in the number of heads appearing each time the exper-
iment is carried out. This leads to the definition of the r.v. X by: X(s) = # of
H’s in s. Thus, X(HHH) = 3, X(HHT) = X(HTH) = X(THH) = 2, X(HTT) =
X(THT) = X(TTH) = 1, and X(TTT) = 0, so that X(S) = {0, 1, 2, 3}. The nota-
tion (X ≤ 1) stands for the event {s ∈ S; X(s) ≤ 1} = {TTT, HTT, THT, TTH}.
In the general case and for B ⊆ , the notation (X ∈ B) stands for the event
A in the sample space S defined by: A = {s ∈ S; X(s) ∈ B}. It is also denoted
by X−1
(B).
In reference to Example 27, a r.v. X of interest may be defined by X(s) =
sum of the numbers in the pair s. Thus, X((1, 1)) = 2, X((1, 2)) = X((2, 1)) =
3, . . . , X((6, 6)) = 12, and X(S) = {2, 3, . . . , 12}. Also, X−1
({7}) = {s ∈ S;
X(s) = 7} = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}. Similarly for Examples
28–31.
In reference to Example 32, a natural r.v. X is defined to denote the num-
ber of tosses needed until the first head occurs. Thus, X(H) = 1, X(T H) =
2, . . . , X(T . . . T

n−1
H) = n, . . . , so that X(S) = {1, 2, . . .}. Also, (X 5) =
(X ≥ 6) = {TTTTTH, TTTTTTH, . . .}.
In reference to Example 33, an obvious r.v. X is: X(s) = s, s = 0, 1, . . . ,
and similarly for Examples 34–35.
In reference to Example 36, a r.v. X of interest is X(s) = s, s ∈ S, and
similarly for Examples 37–40.
Also, in reference to Example 5, an obvious r.v. X may be defined as fol-
lows: X(s) = # of S’s in s. Then, clearly, X(S) = {0, 1, . . . , n}. Similarly for
Examples 6–10.
In reference to Example 11, a r.v. X may be defined thus: X(s) = the position
of M in s. Then, clearly, X(S) = {1, 2, . . .}.
In reference to Example 18, the r.v.’s of obvious interests are: XA = # of
those persons, out of n, having blood type A, and similarly for XB, XAB, XO.
Similarly for Examples 19 and 20.

Exercises 21
From the preceding examples, two kinds of r.v.’s emerge: random vari-
ables which take on countably many values, such as those defined in conjunc-
tion with Examples 26–31 and 32–35, and r.v.’s which take on all values in a
nondegenerate (finite or not) interval in . Such are r.v.’s defined in conjunc-
tion with Examples 36–40. Random variables of the former kind are called
discrete r.v.’s (or r.v.’s of the discrete type), and r.v.’s of the latter type are
called continuous r.v.’s (or r.v.’s of the continuous type).
More generally, a r.v. X is called discrete (or of the discrete type), if X takes
on countably many values; i.e., either finitely many values such as x1, . . . , xn,
or countably infinite many values such as x0, x1, . . . or x1, x2, . . . . On the other
hand, X is called continuous (or of the continuous type) if X takes all values
in a proper interval I ⊆ . Although there are other kinds of r.v.’s, in this book
we will restrict ourselves to discrete and continuous r.v.’s as just defined.
The study of r.v.’s is one of the main objectives of this book.
Exercises
3.1 In reference to Exercise 2.1, define the r.v.’s Xi, i = 1, 2, 3 as follows:
Xi = # of passengers delivered to hotel Hi.
Determine the values of each Xi, i = 1, 2, 3, and specify the values of the
sum X1 + X2 + X3.
3.2 In reference to Exercise 2.2, define the r.v.’s X and Y as follows: X = # of
red balls dispensed, Y = # of balls other than red dispensed.
Determine the values of X and Y, and specify the values of the sum X +Y.
3.3 In reference to Exercise 2.5, define the r.v.’s X and Y as follows: X = # of
pumps in use in station I, Y = # of pumps in use in station II.
Determine the values of X and Y, and also of the sum X + Y.
3.4 In reference to Exercise 2.7, define the r.v. X by: X((x, y)) = x + y.
Determine the values of X, as well as the following events: (X ≤ 2),
(3 X ≤ 5), (X 6).
3.5 Consider a year with 365 days, which are numbered serially from 1 to 365.
Ten of those numbers are chosen at random and without replacement,
and let X be the r.v. denoting the largest number drawn.
Determine the values of X.
3.6 A four-sided die has the numbers 1 through 4 written on its sides, one on
each side. If the die is rolled twice:
(i) Write out a suitable sample space S.
(ii) If X is the r.v. denoting the sum of numbers appearing, determine the
values of X.
(iii) Determine the events: (X ≤ 3), (2 ≤ X 5), (X 8).
3.7 From a certain target population, n individuals are chosen at random
and their blood types are determined. Let X1, X2, X3, and X4 be the r.v.’s

denoting the number of individuals having blood types A, B, AB, and O,
respectively.
Determine the values of each one of these r.v.’s, as well as the values of
the sum X1 + X2 + X3 + X4.
3.8 A bus is expected to arrive at a speciﬁed bus stop any time between 8:00
and 8:15 a.m., and let X be the r.v. denoting the actual time of arrival of
the bus.
(i) Determine the suitable sample space S for the experiment of observ-
ing the arrival of the bus.
(ii) What are the values of the r.v. X?
(iii) Determine the event: “The bus arrives within 5 minutes before the
expiration of the expected time of arrival.”

Chapter 2
The Concept
of Probability
and Basic Results
This chapter consists of five sections. The first section is devoted to the def-
inition of the concept of probability. We start with the simplest case, where
complete symmetry occurs, proceed with the definition by means of relative
frequency, and conclude with the axiomatic definition of probability. The defin-
ing properties of probability are illustrated by way of examples. Also, a number
of basic properties, resulting from the definition, are stated and justified. Some
of them are illustrated by means of examples. The section is concluded with
two theorems, which are stated but not proved.
In the second section, the distribution of a r.v. is introduced. Also, the
distribution function and the probability density function of a r.v. are defined,
and we explain how they determine the distribution of the r.v.
The concept of the conditional probability of an event, given another event,
is taken up in the following section. Its definition is given, and its significance
is demonstrated through a number of examples. This section is concluded
with three theorems, formulated in terms of conditional probabilities. Through
these theorems, conditional probabilities greatly simplify calculation of other-
wise complicated probabilities.
In the fourth section, the independence of two events is defined, and we
also indicate how it carries over to any finite number of events. A result
(Theorem 6) is stated which is often used by many authors without its use
even being acknowledged. The section is concluded with an indication of how
independence extends to random experiments. The definition of independence
of r.v.’s is deferred to another chapter (Chapter 5).
In the final section of the chapter, the so-called fundamental principle
of counting is discussed; combinations and permutations are then obtained
as applications of this principle. Several illustrative examples are also
provided.
23

24 Chapter 2 The Concept of Probability and Basic Results
2.1 Definition of Probability and Some Basic Results
When a random experiment is entertained, one of the first questions which
arise is, what is the probability that a certain event occurs? For instance, in
reference to Example 26 in Chapter 1, one may ask: What is the probability that
exactly one head occurs; in other words, what is the probability of the event
B = {HTT, T HT, TTH}? The answer to this question is almost automatic and
is 3/8. The relevant reasoning goes like this: Assuming that the three coins are
balanced, the probability of each one of the 8 outcomes, considered as simple
events, must be 1/8. Since the event B consists of 3 sample points, it can occur
in 3 different ways, and hence its probability must be 3/8.
This is exactly the intuitive reasoning employed in defining the concept
of probability when two requirements are met: First, the sample space S has
finitely many outcomes, S = {s1, . . . , sn}, say, and second, each one of these
outcomes is “equally likely” to occur, has the same chance of appearing, when-
ever the relevant random experiment is carried out. This reasoning is based
on the underlying symmetry. Thus, one is led to stipulating that each one of
the (simple) events {si}, i = 1, . . . , n has probability 1/n. Then the next step,
that of defining the probability of a composite event A, is simple; if A consists
of m sample points, A = {si1
, . . . , sim
}, say (1 ≤ m ≤ n) (or none at all, in
which case m = 0), then the probability of A must be m/n. The notation used
is: P({s1}) = · · · = P({sn}) = 1
n
and P(A) = m
n
. Actually, this is the so-called
classical definition of probability. That is,
CLASSICAL DEFINITION OF PROBABILITY Let S be a sample space, associ-
ated with a certain random experiment and consisting of finitely many sample
points n, say, each of which is equally likely to occur whenever the random
experiment is carried out. Then the probability of any event A, consisting of
msample points (0 ≤ m ≤ n), is given by P(A) = m
n
.
In reference to Example 26 in Chapter 1, P(A) = 4
8
= 1
2
= 0.5. In Example
27 (when the two dice are unbiased), P(X = 7) = 6
36
= 1
6
0.167, where
the r.v. X and the event (X = 7) are defined in Section 1.3. In Example 29,
when the balls in the urn are thoroughly mixed, we may assume that all of the
(m+n)(m+n−1) pairs are equally likely to be selected. Then, since the event
A occurs in 20 different ways, P(A) = 20
(m+ n)(m+ n− 1)
. For m = 3 and n = 5,
this probability is P(A) = 20
56
= 5
14
0.357.
From the preceding (classical) definition of probability, the following
simple properties are immediate: For any event A, P(A) ≥ 0; P(S) = 1; if two
events A1 and A2 are disjoint (A1 ∩ A2 = ∅), then P(A1 ∪ A2) = P(A1)+ P(A2).
This is so because, if A1 = {si1
, . . . , sik
}, A2 = {sj1
, . . . , sj
}, where all si1
, . . . , sik
are distinct from all sj1
, . . . , sj
, then A1 ∪ A2 = {si1
, . . . , sik
sj1
, . . . , sj
} and
P(A1 ∪ A2) = k+
n
= k
n
+
n
= P(A1) + P(A2).
In many cases, the stipulations made in defining the probability as above
are not met, either because S has not finitely many points (as is the case in
Examples 32, 33–35 (by replacing C and M by ∞), and 36–40 in Chapter 1), or

2.1 Definition of Probability and Some Basic Results 25
because the (finitely many outcomes) are not equally likely. This happens, for
instance, in Example 26 when the coins are not balanced and in Example 27
when the dice are biased. Strictly speaking, it also happens in Example 30. In
situations like this, the way out is provided by the so-called relative frequency
definition of probability. Specifically, suppose a random experiment is carried
out a large number of times N, and let N(A) be the frequency of an event A,
the number of times A occurs (out of N ). Then the relative frequency of A
is N(A)
N
. Next, suppose that, as N → ∞, the relative frequencies N(A)
N
oscillate
around some number (necessarily between 0 and 1). More precisely, suppose
that N(A)
N
converges, as N → ∞, to some number. Then this number is called
the probability of A and is denoted by P(A). That is, P(A) = limN→∞
N(A)
N
.
(It will be seen later in this book that the assumption of convergence of the
relative frequencies N(A)/N is justified subject to some qualifications.) To
summarize,
RELATIVE FREQUENCY DEFINITION OF PROBABILITY Let N(A) be the num-
ber of times an event A occurs in N repetitions of a random experiment, and
assume that the relative frequency of A, N(A)
N
, converges to a limit as N → ∞.
This limit is denoted by P(A) and is called the probability of A.
At this point, it is to be observed that empirical data show that the relative
frequency definition of probability and the classical definition of probability
agree in the framework in which the classical definition applies.
From the relative frequency definition of probability and the usual proper-
ties of limits, it is immediate that: P(A) ≥ 0 for every event A; P(S) = 1; and
for A1, A2 with A1 ∩ A2 = ∅,
P(A1 ∪ A2) = lim
N→∞
N(A1 ∪ A2)
N
= lim
N→∞

N(A1)
N
+
N(A2)
N

= lim
N→∞
N(A1)
N
+ lim
N→∞
N(A2)
N
= P(A1) + P(A2);
that is, P(A1 ∪ A2) = P(A1) + P(A2), provided A1 ∩ A2 = ∅. These three
properties were also seen to be true in the classical definition of probabil-
ity. Furthermore, it is immediate that under either definition of probability,
P(A1 ∪ . . . ∪ Ak) = P(A1) + · · · + P(Ak), provided the events are pairwise
disjoint; Ai ∩ Aj = ∅, i = j.
The above two definitions of probability certainly give substance to the
concept of probability in a way consonant with our intuition about what prob-
ability should be. However, for the purpose of cultivating the concept and
deriving deep probabilistic results, one must define the concept of probability
in terms of some basic properties, which would not contradict what we have
seen so far. This line of thought leads to the so-called axiomatic definition of
probability due to Kolmogorov.
AXIOMATIC DEFINITION OF PROBABILITY Probability is a function, denoted
by P, defined for each event of a sample space S, taking on values in the real

line , and satisfying the following three properties:
(P1) P(A) ≥ 0 for every event A (nonnegativity of P).
(P2) P(S) = 1 (P is normed).
(P3) For countably infinite many pairwise disjoint events Ai, i = 1, 2, . . . , Ai ∩
Aj = ∅, i = j, it holds
P(A1 ∪ A2 ∪ . . .) = P(A1) + P(A2) + · · · ; or P

∞

i=1
Ai

=
∞

i=1
P(Ai)
(sigma-additivity (σ-additivity) of P).
COMMENTS ON THE AXIOMATIC DEFINITION
1) Properties (P1) and (P2) are the same as the ones we have seen earlier,
whereas property (P3) is new. What we have seen above was its so-called
finitely-additive version; that is, P(
n
i=1 Ai) =
n
i=1 P(Ai), provided Ai ∩
Aj = ∅, i = j. It will be seen below that finite-additivity is implied by
σ-additivity but not the other way around. Thus, if we are to talk about the
probability of the union of countably infinite many pairwise disjoint events,
property (P3) must be stipulated. Furthermore, the need for such a union
of events is illustrated as follows: In reference to Example 32, calculate the
probability that the first head does not occur before the nth tossing. By
setting Ai = {T . . . T

i
H}, i = n, n + 1, . . . , what we are actually after here
is P(An ∪ An+1 ∪ . . .) with Ai ∩ Aj = ∅, i = j, i and j ≥ n.
2) Property (P3) is superfluous (reduced to finite-additivity) when the sample
space S is finite, which implies that the total number of events is finite.
3) Finite-additivity is implied by additivity for two events, P(A1 ∪ A2) =
P(A1) + P(A2), A1 ∩ A2 = ∅, by way of induction.
Here are two examples in calculating probabilities.
EXAMPLE 1 In reference to Example 1 in Chapter 1, take n = 58, and suppose we have the
following configuration:
BARIUM
HIGH LOW
Mercury Mercury
Arsenic High Low High Low
High 1 3 5 9
Low 4 8 10 18
Calculate the probabilities mentioned in (i) (a)–(d).
DISCUSSION For simplicity, denote by Bh the event that the site selected
has a high barium concentration, and likewise for other events figuring below.
Then:

(i)(a) Bh = (Ah ∩ Bh ∩ Mh)∪(Ah ∩ Bh ∩ M)∪(A ∩ Bh ∩ Mh)∪(A ∩ Bh ∩ M)
and the events on the right-hand side are pairwise disjoint. Therefore
(by the following basic property 2 in Subsection 2.1.1):
P(Bh) = P(Ah ∩ Bh ∩ Mh) + P(Ah ∩ Bh ∩ M)
+ P(A ∩ Bh ∩ Mh) + P(A ∩ Bh ∩ M)
=
1
58
+
3
58
+
4
58
+
8
58
=
16
58
=
8
29
0.276.
(i)(b) Here P(Mh ∩ A ∩ B) = P(A ∩ B ∩ Mh) = 10
58
= 5
29
0.172.
(i)(c) Here the required probability is as in (a):
P(Ah∩Bh∩M) + P(Ah∩B∩Mh) + P(A∩Bh∩Mh) =
12
58
=
6
29
0.207.
(i)(d) As above,
P(Ah ∩ B ∩ M) + P(A ∩ Bh ∩ M) + P(A ∩ B ∩ Mh) =
27
58
0.466.
EXAMPLE 2 In ranking five horses in a horse race (Example 31 in Chapter 1), calculate the
probability that horse #3 terminates at least second.
DISCUSSION Let Ai be the event that horse #3 terminates in the ith posi-
tion, i = 1, . . . , 5. Then the required event is A1 ∪ A2, where A1, A2 are disjoint.
Thus,
P(A1 ∪ A2) = P(A1) + P(A2) =
24
120
+
24
120
=
2
5
= 0.4.
EXAMPLE 3 In tossing a coin repeatedly until H appears for the first time (Example 32 in
Chapter 1), suppose that P{T . . . T

i−1
H} = P(Ai) = qi−1
p for some 0 p 1
and q = 1 − p (in anticipation of Definition 3 in Section 2.4). Then
P

∞

i=n
Ai

=
∞

i=n
P(Ai) =
∞

i=n
qi−1
p = p
∞

i=n
qi−1
= p
qn−1
1 − q
= p
qn−1
p
= qn−1
.
For instance, for p = 1/2 and n = 3, this probability is 1
4
= 0.25. That is, when
tossing a fair coin, the probability that the first head does not appear either
the first or the second time (and therefore it appears either the third time
or the fourth time etc.) is 0.25. For n = 10, this probability is approximately
0.00195 0.002.
Next, we present some basic results following immediately from the defining
properties of the probability. First, we proceed with their listing and then with
their justification.

2.1.1 Some Basic Properties of a Probability Function
1. P(∅) = 0.
2. For any pairwise disjoint events A1, . . . , An, P(
n
i=1 Ai) =
n
i=1 P(Ai).
3. For any event A, P(Ac
) = 1 − P(A).
4. A1 ⊆ A2 implies P(A1) ≤ P(A2) and P(A2 − A1) = P(A2) − P(A1).
5. 0 ≤ P(A) ≤ 1 for every event A.
6. (i) For any two events A1 and A2:
P(A1 ∪ A2) = P(A1) + P(A2) − P(A1 ∩ A2).
(ii) For any three events A1, A2, and A3:
P(A1 ∪ A2 ∪ A3) = P(A1) + P(A2) + P(A3) − [P(A1 ∩ A2)
+ P(A1 ∩ A3) + P(A2 ∩ A3)] + P(A1 ∩ A2 ∩ A3).
7. For any events A1, A2, . . . , P(
∞
i=1 Ai) ≤
∞
i=1 P(Ai) (σ-sub-additivity),
and P(
n
i=1 Ai) ≤
n
i=1 P(Ai) (ﬁnite-sub-additivity).
2.1.2 Justification
1. From the obvious fact that S = S ∪ ∅ ∪ ∅ ∪ . . . and property (P3),
P(S) = P(S ∪ ∅ ∪ ∅ ∪ . . .) = P(S) + P(∅) + P(∅) + · · ·
or P(∅) + P(∅) + · · · = 0. By (P1), this can only happen when P(∅) = 0.
Of course, that the impossible event has probability 0 does not come as a
surprise. Any reasonable deﬁnition of probability should imply it.
2. Take Ai = ∅ for i ≥ n+ 1, consider the following obvious relation, and use
(P3) and #1 to obtain:
P

n

i=1
Ai

= P

∞

i=1
Ai

=
∞

i=1
P(Ai) =
n

i=1
P(Ai).
3. From (P2) and #2, P(A ∪ Ac
) = P(S) = 1 or P(A) + P(Ac
) = 1, so that
P(Ac
) = 1 − P(A).
4. The relation A1 ⊆ A2, clearly, implies A2 = A1 ∪ (A2 − A1), so that, by #2,
P(A2) = P(A1) + P(A2 − A1). Solving for P(A2 − A1), we obtain P(A2 −
A1) = P(A2) − P(A1), so that, by (P1), P(A1) ≤ P(A2).
At this point it must be pointed out that P(A2 − A1) need not be P(A2) −
P(A1), if A1 is not contained in A2.
5. Clearly, ∅ ⊆ A ⊆ S for any event A. Then (P1), #1 and #4 give: 0 = P(∅) ≤
P(A) ≤ P(S) = 1.
6. (i) It is clear (by means of a Venn diagram, for example) that
A1 ∪ A2 = A1 ∪ A2 ∩ Ac
1 = A1 ∪ (A2 − A1 ∩ A2).
Then, by means of #2 and #4:
P(A1 ∪ A2) = P(A1) + P(A2 − A1 ∩ A2) = P(A1) + P(A2) − P(A1 ∩ A2).

(ii) Apply part (i) to obtain:
P(A1 ∪ A2 ∪ A3) = P[(A1 ∪ A2) ∪ A3] = P(A1 ∪ A2) + P(A3)
− P[(A1 ∪ A2) ∩ A3]
= P(A1) + P(A2) − P(A1 ∩ A2) + P(A3)
− P[(A1 ∩ A3) ∪ (A2 ∩ A3)]
= P(A1) + P(A2) + P(A3) − P(A1 ∩ A2)
− [P(A1 ∩ A3) + P(A2 ∩ A3) − P(A1 ∩ A2 ∩ A3)]
= P(A1) + P(A2) + P(A3) − P(A1 ∩ A2) − P(A1 ∩ A3)
− P(A2 ∩ A3) + P(A1 ∩ A2 ∩ A3).
7. By the identity in Section 2 of Chapter 1 and (P3):
P

∞

i=1
Ai

= P

A1 ∪ Ac
1 ∩ A2 ∪ . . . ∪ Ac
1 ∩ . . . ∩ Ac
n−1 ∩ An ∪ . . .

= P(A1) + P Ac
1 ∩ A2 + · · · + P Ac
1 ∩ . . . ∩ Ac
n−1 ∩ An + · · ·
≤ P(A1) + P(A2) + · · · + P(An) + · · · (by #4).
For the ﬁnite case:
P

n

i=1
Ai

= P

A1 ∪ Ac
1 ∩ A2 ∪ . . . ∪ Ac
1 ∩ . . . ∩ Ac
n−1 ∩ An

= P(A1) + P Ac
1 ∩ A2 + · · · + P Ac
1 ∩ . . . ∩ Ac
n−1 ∩ An
≤ P(A1) + P(A2) + · · · + P(An).
Next, some examples are presented to illustrate some of the properties
#1–#7.
EXAMPLE 4 (i) For two events A and B, suppose that P(A) = 0.3, P(B) = 0.5, and P(A∪
B) = 0.6. Calculate P(A ∩ B).
(ii) If P(A) = 0.6, P(B) = 0.3, P(A ∩ Bc
) = 0.4, and B ⊂ C, calculate P(A ∪
Bc
∪ Cc
).
DISCUSSION
(i) From P(A ∪ B) = P(A) + P(B) − P(A ∩ B), we get P(A ∩ B) = P(A) +
P(B) − P(A ∪ B) = 0.3 + 0.5 − 0.6 = 0.2.
(ii) The relation B ⊂ C implies Cc
⊂ Bc
and hence A ∪ Bc
∪ Cc
= A ∪ Bc
.
Then P(A ∪ Bc
∪ Cc
) = P(A ∪ Bc
) = P(A) + P(Bc
) − P(A ∩ Bc
) =
0.6 + (1 − 0.3) − 0.4 = 0.9.
EXAMPLE 5 Let A and B be the respective events that two contracts I and II, say, are
completed by certain deadlines, and suppose that: P(at least one contract
is completed by its deadline) = 0.9 and P(both contracts are completed by

their deadlines) = 0.5. Calculate the probability: P(exactly one contract is
completed by its deadline).
DISCUSSION Theassumptionsmadearetranslatedasfollows: P(A∪ B) =
0.9 and P(A∩ B) = 0.5. What we wish to calculate is: P((A∩ Bc
)∪(Ac
∩ B)) =
P(A ∩ Bc
) + P(Ac
∩ B). Clearly, A = (A ∩ B) ∪ (A ∩ Bc
) and B = (A ∩ B) ∪
(Ac
∩ B), so that P(A) = P(A ∩ B) + P(A ∩ Bc
) and P(B) = P(A ∩ B) +
P(Ac
∩ B). Hence, P(A ∩ Bc
) = P(A) − P(A ∩ B) and P(Ac
∩ B) = P(B) −
P(A ∩ B). Then P(A ∩ Bc
) + P(Ac
∩ B) = P(A) + P(B) − 2(A ∩ B) =
[P(A)+ P(B)− P(A∩ B)]− P(A∩ B) = P(A∪ B)− P(A∩ B) = 0.9−0.5 = 0.4.
EXAMPLE 6 (i) For three events A, B, and C, suppose that P(A ∩ B) = P(A ∩ C) and
P(B ∩ C) = 0. Then show that P(A ∪ B ∪ C) = P(A) + P(B) + P(C) −
2P(A ∩ B).
(ii) For any two events A and B, show that P(Ac
∩ Bc
) = 1 − P(A) − P(B) +
P(A ∩ B).
DISCUSSION
(i) We have P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) −
P(B ∩ C) + P(A∩ B ∩ C). But A∩ B ∩ C ⊂ B ∩ C, so that P(A∩ B ∩ C) ≤
P(B ∩ C) = 0, and therefore P(A ∪ B ∪ C) = P(A) + P(B) + P(C) −
2P(A ∩ B).
(ii) Indeed, P(Ac
∩ Bc
) = P((A ∪ B)c
) = 1 − P(A ∪ B) = 1 − P(A) − P(B) +
P(A ∩ B).
EXAMPLE 7 In ranking five horses in a horse race (Example 31 in Chapter 1), what is the
probability that horse #3 will terminate either first or second or third?
DISCUSSION Denote by B the required event and let Ai = “horse #3
terminates in the ith place,” i = 1, 2, 3. Then the events A1, A2, A3 are pairwise
disjoint, and therefore
P(B) = P(A1 ∪ A2 ∪ A3) = P(A1) + P(A2) + P(A3).
But P(A1) = P(A2) = P(A3) = 24
120
= 0.2, so that P(B) = 0.6.
EXAMPLE 8 Consider a well-shuffled deck of 52 cards (Example 28 in Chapter 1), and
suppose we draw at random three cards. What is the probability that at least
one is an ace?
DISCUSSION Let A be the required event, and let Ai be defined by: Ai =
“exactly i cards are aces,” i = 0, 1, 2, 3. Then, clearly, P(A) = P(A1 ∪ A2 ∪
A3). Instead, we may choose to calculate P(A) through P(Ac
) = 1 − P(A0),
where
P(A0) =
48
3
52
3
=
48 × 47 × 46
52 × 51 × 50
=
4,324
5,525
, so that P(A) =
1,201
5,525
0.217.

EXAMPLE 9 Refer to Example 3 in Chapter 1 and let C1, C2, C3 be defined by: C1 = “both
S1 and S2 work,” C2 = “S5 works,” C3 = “both S3 and S4 work,” and let
C = “current is transferred from point A to point B.” Then P(C) = P(C1 ∪
C2 ∪ C3). At this point (in anticipation of Definition 3 in Section 2.4; see also
Exercise 4.14 in this chapter), suppose that:
P(C1) = p1 p2, P(C2) = p5, P(C3) = p3 p4,
P(C1 ∩ C2) = p1 p2 p5, P(C1 ∩ C3) = p1 p2 p3 p4,
P(C2 ∩ C3) = p3 p4 p5, P(C1 ∩ C2 ∩ C3) = p1 p2 p3 p4 p5.
Then:
P(C) = p1 p2 + p5 + p3 p4 − p1 p2 p5 − p1 p2 p3 p4 − p3 p4 p5 + p1 p2 p3 p4 p5.
For example, for p1 = p2 = p3 = 0.9, we obtain
P(C) = 0.9 + 2(0.9)2
− 2(0.9)3
− (0.9)4
+ (0.9)5
0.996.
This section is concluded with two very useful results stated as theorems.
The first is a generalization of property #6 to more than three events, and the
second is akin to the concept of continuity of a function as it applies to a
probability function.
THEOREM 1
The probability of the union of any n events, A1, . . . , An, is given by:
P

n

j=1
Aj

=
n

j=1
P(Aj) −

1≤ j1 j2≤n
P(Aj1
∩ Aj2
)
+

1≤ j1 j2 j3≤n
P(Aj1
∩ Aj2
∩ Aj3
) − · · ·
+ (−1)n+1
P(A1 ∩ . . . ∩ An).
Although its proof (which is by induction) will not be presented, the pattern
of the right-hand side above follows that of property #6(i) and it is clear. First,
sumuptheprobabilitiesoftheindividualevents,thensubtracttheprobabilities
of the intersections of the events, taken two at a time (in the ascending order
of indices), then add the probabilities of the intersections of the events, taken
three at a time as before, and continue like this until you add or subtract
(depending on n) the probability of the intersection of all n events.
Recall that, if A1 ⊆ A2 ⊆ . . . , then limn→∞ An =
∞
n=1 An, and if A1 ⊇ A2 ⊇
. . . , then limn→∞ An = ∞
n=1 An.
THEOREM 2
For any monotone sequence of events {An}, n ≥ 1, it holds P(limn→∞
An) = limn→∞ P(An).

This theorem will be employed in many instances, and its use will be then
pointed out.
Exercises
1.1 If P(A) = 0.4, P(B) = 0.6, and P(A ∪ B) = 0.7, calculate P(A ∩ B).
1.2 If for two events A and B, it so happens that P(A) = 3
4
and P(B) = 3
8
,
show that:
P(A ∪ B) ≥
3
4
and
1
8
≤ P(A ∩ B) ≤
3
8
.
1.3 If for the events A, B, andC, it so happens that P(A) = P(B) = P(C) = 1,
then show that:
P(A ∩ B) = P(A ∩ C) = P(B ∩ C) = P(A ∩ B ∩ C) = 1.
1.4 If the events A, B, and C are related as follows: A ⊂ B ⊂ C and P(A) =
1
4
, P(B) = 5
12
, and P(C) = 7
12
, compute the probabilities of the following
events:
Ac
∩ B, Ac
∩ C, Bc
∩ C, A ∩ Bc
∩ Cc
, Ac
∩ Bc
∩ Cc
.
1.5 Let S be the set of all outcomes when flipping a fair coin four times, so
that all 16 outcomes are equally likely. Define the events A and B by:
A = {s ∈ S; s contains more Ts than Hs},
B = {s ∈ S; any T in s precedes every H in s}.
Compute the probabilities P(A), P(B).
1.6 Let S = {x integer; 1 ≤ x ≤ 200}, and define the events A, B, and C as
follows:
A = {x ∈ S; x is divisible by 7},
B = {x ∈ S; x = 3n + 10, for some positive integer n},
C = {x ∈ S; x2
+ 1 ≤ 375}.
Calculate the probabilities P(A), P(B), and P(C).
1.7 If two fair dice are rolled once, what is the probability that the total
number of spots shown is:
(i) Equal to 5?
(ii) Divisible by 3?
1.8 Students in a certain college subscribe to three news magazines A, B,
and C according to the following proportions:
A : 20%, B : 15%, C : 10%,
both A and B : 5%, both A and C : 4%, both B and C : 3%, all three A, B,
and C : 2%.

If a student is chosen at random, what is the probability he/she subscribes
to none of the news magazines?
1.9 A high school senior applies for admissions to two colleges A and B,
and suppose that: P(admitted at A) = p1, P(rejected by B) = p2, and
P(rejected by at least one, A or B) = p3.
(i) Calculate the probability that the student is admitted by at least one
college.
(ii) Find the numerical value of the probability in part (i), if p1 = 0.6, p2 =
0.2, and p3 = 0.3.
1.10 An airport limousine service has two vans, the smaller of which can carry
6 passengers and the larger 9 passengers. Let x and y be the respective
numbers of passengers carried by the smaller and the larger van in a given
trip, so that a suitable sample space S is given by:
S = {(x, y); x = 0, . . . , 6 and y = 0, 1, . . . , 9}.
Also, suppose that, for all values of x and y, the probabilities P({(x, y)})
are equal. Finally, deﬁne the events A, B, and C as follows:
A = “the two vans together carry either 4 or 6 or 10 passengers,”
B = “the larger van carries twice as many passengers as the smaller
van,”
C = “the two vans carry different numbers of passengers.”
Calculate the probabilities: P(A), P(B), and P(C).
1.11 In the sample space S = (0, ∞), consider the events An = (0, 1 − 2
n
),
n = 1, 2, . . . , A = (0, 1), and suppose that P(An) = 2n−1
4n
.
(i) Show that the sequence {An} is increasing and that limn→∞ An =
∞
n=1 An = A.
(ii) Use part (i) and the appropriate theorem (cite it!) in order to calculate
the probability P(A).
2.2 Distribution of a Random Variable
For a r.v. X, deﬁne the set function PX(B) = P(X ∈ B). Then PX is a prob-
ability function because: PX(B) ≥ 0 for all B, PX() = P(X ∈ ) = 1, and, if
Bj, j = 1, 2, . . . are pairwise disjoint then, clearly, (X ∈ Bj), j ≥ 1, are also
pairwise disjoint and X ∈ (
∞
j=1 Bj) =
∞
j=1(X ∈ Bj). Therefore
PX

∞

j=1
Bj

= P

X ∈

∞

j=1
Bj

= P

∞

j=1
(X ∈ Bj)

=
∞

j=1
P(X ∈ Bj) =
∞

j=1
PX(Bj).

The probability function PX is called the probability distribution of the r.v.
X. Its significance is extremely important because it tells us the probability that
X takes values in any given set B. Indeed, much of probability and statistics
revolves around the distribution of r.v.’s in which we have an interest.
By selecting B to be (−∞, x], x ∈ , we have PX(B) = P(X ∈ (−∞, x]) =
P(X ≤ x). In effect, we define a point function which we denote by FX; that is,
FX(x) = P(X ≤ x), x ∈ . The function FX is called the distribution function
(d.f.) of X. Clearly, if we know PX, then we certainly know FX. Somewhat un-
expectedly, the converse is also true. Namely, if we know the (relatively “few”)
probabilities FX(x), x ∈ , then we can determine precisely all probabilities
PX(B) for B subset of . This converse is a deep theorem in probability that
we cannot deal with here. It is, nevertheless, the reason for which it is the d.f.
FX we deal with, a familiar point function for which so many calculus results
hold, rather than the unfamiliar set function PX.
Clearly, the expressions FX(+∞) and FX(−∞) have no meaning because
+∞ and −∞ are not real numbers. They are defined as follows:
FX(+∞) = lim
n→∞
FX(xn), xn ↑ ∞ and FX(−∞) = lim
n→∞
FX(yn), yn ↓ −∞.
These limits exist because x y implies (−∞, x] ⊂ (−∞, y] and hence
PX((−∞, x]) = FX(x) ≤ FX(y) = PX((−∞, y]).
The d.f. of a r.v. X has the following basic properties:
1. 0 ≤ FX(x) ≤ 1 for all x ∈ ;
2. FX is a nondecreasing function;
3. FX is continuous from the right;
4. FX(+∞) = 1, FX(−∞) = 0.
The first and the second properties are immediate from the definition of the
d.f.; the third follows by Theorem 2, by taking xn ↓ x; so does the fourth, by
taking xn ↑ +∞, which implies (−∞, xn] ↑ , and yn ↓ −∞, which implies
(−∞, yn] ↓ ∅. Figures 2.1 and 2.2 show the graphs of the d.f.’s of some typical
cases.
Now, suppose that the r.v. X is discrete and takes on the values xj, j =
1, 2, . . . , n. Take b = {xj} and on the set {x1, x2, . . . , xn} define the function fX
as follows: fX(xj) = PX({xj}). Next, extend fX over the entire by setting
0.20
0.40
0.60
0.80
1.00
0
F(x)
x
0.20
0.40
0.60
0.80
1.00
0
F(x)
x
(b) Poisson for l = 2.
(a) Binomial for n = 6, p =
1
–
4
.
Figure 2.1
Examples of Graphs of
d.f.’s

Φ(x)
1.0
0.5
−2 −1 0
(d) N(0, 1).
1 2
x
1.0
0
b
a
F(x)
x
(c) U(a, b ).
0
1
x a
x b
a ≤ x ≤ b.
x − a
b − a
Here F(x) =
Figure 2.2
Examples of Graphs of
d.f.’s
fX(x) = 0 for x = xj, j = 1, 2, . . . , n. Then fX(x) ≥ 0 for all x, and it is
clear that P(X ∈ B) =

xj∈B fX(xj) for B ⊆ . In particular,
n
j=1 fX(xj) =

xj∈ fX(xj) = P(X ∈ ) = 1. The function fX just deﬁned is called the
probability density function (p.d.f.) of the r.v. X. By selecting B = (−∞, x]
for some x ∈ , we have FX(x) =

xj≤x fX(xj). Furthermore, if we assume at
this point that x1 x2 · · · xn, it is clear that
fX(xj) = FX(xj) − FX(xj−1), j = 2, 3, . . . , n and fX(x1) = FX(x1);
we may also allow j to take the value 1 above by setting FX(x0) = 0. Likewise
if X takes the values xj, j = 1, 2, . . . These two relations state that, in the
case that X is a discrete r.v. as above, either one of the FX of fX speciﬁes
uniquely the other. Setting FX(xj−) for the limit from the left (left-limit) of
FX at xj, FX(xj−) = lim FX(x) as x ↑ xj, we see that FX(xj) − FX(xj−1) =
FX(xj) − FX(xj−), so that fX(xj) = FX(xj) − FX(xj−). In other words, the
value of fX at xj is the size of the jump of FX at the point xj. These points
are illustrated quite clearly in Figure 2.3. For a numerical example (associated
with Figure 2.3), let the r.v. X take on the values: −14, −6, 5, 9, and 24 with
respective probabilities: 0.17, 0.28, 0.22, 0.22, and 0.11.
−14 −6 0 5 9 24
x
F(x)
0.17
0.28
0.22
0.22
0.11
1
Figure 2.3

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By a process of natural selection native shipping in China and Japan
has been extensively superseded by foreign, and an immense
dislocation of capital has in consequence taken place. The effect of
this has been severely felt on the China coast, especially in such
large shipping ports as Taku, Shanghai, and Ningpo, where there
were in former days large and prosperous shipowning communities.
The disturbance has probably been much less marked in Japan,
owing to the greater agility of the people in adapting themselves to
inevitable changes. Certain it is that in both countries there is still a
large junk fleet employed in the coasting trade, being protected
against foreign as well as steam competition by their light draught
and their privilege of trading at ports not opened to foreign trade.
The temptation to evade the prohibition of foreign flags led in former
days to sundry bizarre effects on the coast of China. The natives,
finding it to their advantage to employ foreign vessels, exercised
their ingenuity in making them look like Chinese craft. This would at
first sight appear no easy matter, seeing that the Chinese junks
carried no yards and their hulls were of a construction as different
from that of a modern ship as was possible for two things to be
which were intended for the same purpose. The junks possessed
certain qualities conducive to buoyancy and safety, such as water-
tight bulkheads, which at once strengthened the hull and minimised
the danger of sinking. But their sailing properties, except with the
wind free, were beneath contempt. Their weatherly and seaworthy
qualities commended vessels of foreign construction to the Chinese
traders, while the talisman of the flag was deemed by them a
protection against pirates, and perhaps also, on occasion, against
official inquisition. Probably what on the whole the native owner or
charterer would have preferred was that his ship should pass for
foreign at sea and for native in port. To this end in some cases resort
was had to hermaphrodite rigging, and very generally to two
projecting boards, one on each side of the figurehead, bearing the
staring Chinese eye, such as the junks south of the Yangtze carry.
The open eye on the ship's bow was to enable the Chinese port
officials to close theirs to the unauthorised presence of strangers,

and thus everything was arranged in the manner so dear to the
Chinese character.
In the south of China the advantage of the flag was sought without
the foreign appearance of the vessel. The foreign flag was hoisted
on native-built small craft, a large fleet of which hailed from Macao
under Portuguese colours, and were from time to time guilty of great
irregularities on the coast. The Chinese of Hongkong, British subjects
born and bred, registered their vessels and received colonial sailing
letters, renewable at frequent intervals, as a check on bad behaviour.
With these papers short trips were made along the south coast, and
a local trade was carried on in the estuary of the Canton river. These
vessels of about 100 or 200 tons burthen were called lorchas, of
which we shall hear more in subsequent chapters.

CHAPTER XIII.
THE TRADERS.
I. FOREIGN.
Their relations to their official representatives—And to the trading interests of their
own countries—Their unity—High character—Liberality—Breadth of view.
In the preceding portions of this narrative it has been shown how
much the character of the principal officials on both sides influenced
the progress of events. There was, however, yet another factor
which contributed in a lesser degree and in a different manner to the
general result which ought not to be entirely omitted from
consideration, and that was the personal qualities and traditional
characteristics of the two trading communities, foreign and Chinese.
It was they who created the subject-matter of all foreign relations,
and stood in the breach in all the struggles between foreign and
native officials. It was their persons and their fortunes which were
ever at stake; it was they who first felt the shock of disturbance, and
were the first to reap the fruits of peace.
The relation of the foreign mercantile community to their official
representatives was not always free from friction, because the same
high authority which enjoined on the officials the protection of the
persons and the promotion of the interests of the lay community
empowered them also to rule over these their protégés, and to apply
to them an arbitrary discipline in accordance with what they
conceived to be the exigencies of the time. Duty in such

circumstances must often have assumed a divided aspect, and rules
of action must frequently have been put to a severe strain; nor is it
surprising that, owing to these peculiar relationships, the resident
communities should not have been able on all occasions to see eye
to eye with the agents of their Governments.
In their national and representative character the China merchants
were wont at different crises to have moral burdens laid on them
which did not properly fit their shoulders. They were little affected
by the shallow moralism of the pulpit, which, taken literally, would
have counselled general liquidation and the distribution of the
proceeds among the poor, leaving the common creditor out of
account; but official sermons also were on certain occasions
preached to, or at, the merchants, implying some obligation on their
part to sacrifice individual advantage to the greater good of the
greater number. Were there no other answer to such altruistic
monitions, it would be sufficient to plead that under such theories of
duty commerce could not exist, and its political accessories would
become superfluous. No road to commercial prosperity has been
discovered which could dispense with the prime motive for the
exertion which makes for progress—to wit, individual ambition,
cupidity, or by whatever term we choose to designate the driving
power of the complex machine of civilised life. Mammon is, after all,
a divinity whose worship is as universal as that of Eros, and is
scarcely less essential to the preservation of the race. Nor is it by
collective, but by strictly individual, offerings that these deities are
propitiated, and the high purposes of humanity subserved. It is no
reproach, therefore, to the China merchants that they should have
seized every opportunity for gain, totally irrespective of the general
policy of their country. It was not for them to construe portents, but
to improve the shining hour. And if it should at any time happen that
the action of private persons, impelled by the passion for gain,
embarrassed a diplomatist in his efforts to bring about some grand
international combination, the fault was clearly his who omitted to
take account of the ruling factor in all economic problems. The trade
was not made for Government policy, but the policy for the trade,

whose life-blood was absolute liberty of action and a free course for
individual initiative. The success of British trade as a whole could
only be the aggregate of the separate successes not otherwise
attainable than by each member of the mercantile fraternity
performing his own part with singleness of purpose. Nothing
certainly could ever justify any trader in foregoing a chance of gain
for the sake of an ideal benefit to the community, even if it were
likely to be realised. A distinction must be drawn between the
tradesman and the statesman. Though their functions may
sometimes overlap, their respective duties to the State are of a
different though complementary character.
To the charge which from time to time has been levelled at the
China merchants, that they were too narrow and too selfish, it may
be plausibly replied that, on the contrary, they were if anything too
broad; for their individual interests were not so bound up with
general progress as are the interests of colonists in a new country,
where co-operation is essential. Progress meant, to the China
merchants, the admitting of the flood of competition, which they
were in no condition to meet. The general interests of the country
required the opening of new markets; in a lesser degree the
interests of the manufacturing section required the same thing; but
the interests of the merchants, albeit they appeared to represent
their country and its industries, were in fact opposed to expansion.
Yet so strong in them was the race instinct for progress that their
private advantage has oftentimes actually given way to it, so that we
have seen throughout the developments of foreign intercourse with
China the resident merchants placing themselves in the van in
helping to let loose the avalanche which overwhelmed them and
brought fresh adventurers to occupy the ground.
Nor has the relation of the merchants, even to the operations in
which they were engaged, been always clearly understood. Although
they personified their national trade in the eyes of the world, the
merchants were never anything more than the vehicles for its
distribution, having no interest in its general extension, though a

powerful interest in the increase of their individual share. The
productions which provided the livelihood of many thousands of
people in China, and perhaps of a still larger number in Great Britain
and other manufacturing countries, did not concern them. A
percentage by way of toll on merchandise passing through their
warehouses was the limit of their ambition. A clear distinction should
therefore be drawn between the merchant and the producer or
manufacturer; on which point some observations of Wingrove
Cooke[31] are worth quoting:—
The calculations of the merchants do not extend beyond their own
business. Why should they? Fortunately for himself, the merchant's
optics are those of the lynx rather than those of the eagle. An
extremely far-sighted commercial man must always run risks of
bankruptcy, for the most absolutely certain sequences are often the
most uncertain in point of time. The same writer, however,
comments on the ignorance and narrowness of both British traders
and manufacturers, and their failure to avail themselves of the
opportunities offered to them of exploiting the trading resources of
the Chinese. There is no spirit of inquiry abroad, he says, no
energy at work, no notion of distracting the eye for a moment from
watching those eternal shirtings, no thought whether you cannot
make better shift with some other class of goods. Manchester made
a great blind effort when the ports were opened, and that effort
failed. Since then she has fallen into an apathy, and trusts to the
chapter of accidents. As for the merchants on whom manufacturers
relied to push the sale of their wares, they come out here, he says,
to make fortunes in from five to seven years, not to force English
calicoes up into remote places. Their work is to buy Chinese
produce, but, he goes on, if the English manufacturer wants
extraordinary exertion, carefully collected information, and
persevering up-country enterprise—and this is what he does want—
he must do it himself. The British export trade will not maintain
mercantile houses, but it would pay for travelling agents acting in
immediate connection with the home manufacturers, who should
keep their principals at home well informed, and who should work

their operations through the established houses here. The evil is that
British goods are not brought under the eyes of the Chinaman of the
interior cities.
The inaccuracies of some of these comments need not obscure the
shrewd and prophetic character of the general advice tendered to
the British manufacturers. After an interval of forty years they have
begun to act upon it, and though their progress has as yet been
slow, they are taking to heart another portion of Mr Cooke's advice,
that all dealing with the interior of China is impossible unless your
agents speak the language of the people.
A certain divergence between the official and non-official view of
affairs had begun to show itself in the period before the war. Before
the close of the East India Company's monopoly the independent
merchants perceived that their interests, as well as those of the
Company itself, were prejudiced by the truckling tactics of its agents,
and though few in number, the mercantile community began to give
utterance to their grievances and to show they had a mind of their
own on public commercial policy. As the whole position of foreigners
in China rested on premisses which were essentially false,
disappointment, irritation, and alarm were chronic. Every one
concerned, official and unofficial, was aggrieved thereby, while no
one was disposed to accept blame for the grievance. A tendency to
recrimination was the natural consequence. When their
representatives failed to protect them against the aggressions of the
Chinese the merchants complained, while the officials in their turn
were not indisposed to retort by alleging provocative or injudicious
conduct on the part of the merchants themselves as contributory to
the ever-recurrent difficulties. Through the retrospective vista of two
generations it is easy now to see where both parties were at fault—
the merchants in making too little account of the difficulties under
which their representatives were labouring, and the officials in failing
to perceive that the causes of their disagreements with the Chinese
lay altogether deeper than the casual imprudence of any private
individual, even if that could be established. The despatches of the

earlier superintendents, notably those of Sir George Robinson,
betray a certain jealousy of the political influence supposed to be
wielded by the mercantile community of Canton working through
their associations in England, and the superintendents seemed
therefore concerned to cast discredit on mercantile opinion. It would
have been strange enough, had it been true, that an isolated
community of a hundred individuals should be torn by faction, yet it
is a fact that on their assumed disagreements an argument was
based for invalidating the representations which they occasionally
made to the Home Government. Their views were disparaged, their
motives impugned, and their short-sighted selfishness deplored. The
note struck in 1835 has been maintained with variations down
almost to our own day,—a circumstance which has to be borne in
mind by those who aim at a fair appreciation of British relations with
China during the last sixty years.
Far, however, from being a disunited flock, the mercantile body in
China generally have on the whole been singularly unanimous in
their views of the political transactions with which their interests
were bound up; while as to the old community of Canton, no epithet
could be less appropriate than one which would imply discord.
Concord was the enforced effect of their circumstances. Imprisoned
within a narrow space, surrounded by a hostile people, exposed to a
constant common peril, the foreign residents in Canton were bound
to each other by the mere instinct of self-preservation. They
became, in fact, what Nelson called his captains, a band of
brothers. The exclusion of females up till 1842, and the deterrent
conditions of married life there even under the treaty, made it
essentially a bachelor community, living almost like one family, or as
comrades in a campaign. Of the disinterested hospitality and good-
fellowship which continue to this day, even in the maturity of their
domestic development, to characterise the foreign communities in
China, the germ is doubtless to be discovered in that primitive
society which oscillated between Canton and Macao during the thirty
years which ended in 1856, in which year their factories were for the
last time destroyed, and the old life finally broken up.

But there is something more to be credited to these early residents
than the mutual loyalty prescribed for them by the peculiar
conditions of their life. They exemplified in a special degree the true
temper and feelings of gentlemen,—a moral product with which local
conditions had also, no doubt, something to do. They lived in glass
houses, with open doors; they could by no means get away from
one another, or evade a mutual observation which was constant and
searching. Whatever standards, therefore, were recognised by the
community, the individual members were constrained to live up to
them in a society where words and deeds lay open to the collective
criticism. And the standard was really a high one. Truth, honour,
courage, generosity, nobility, were qualities common to the whole
body; and those who were not so endowed by birthright could not
help assuming the virtue they did not possess, and, through
practice, making it eventually their own. Black sheep there were, no
doubt, but being never whitewashed, they did not infect the flock, as
happens in more advanced communities.
These intimate conditions favouring the formation of character were
powerfully reinforced by the one feature of European life in China
which was external to the residents, their contact with the
surrounding mass of Chinese. The effect of intercourse with so-
called inferior races is a question of much complexity, and large
generalisations on such subjects are unsafe, each case being best
considered on its proper merits. In their intercourse with the
Chinese, certain points stood out like pillars of adamant to fix the
principles by which the foreign residents were obliged to regulate
their bearing towards the natives. In the first place, the strangers
formed units hemmed in and pressed upon by thousands; therefore
they must magnify themselves by maintaining an invincible prestige,
they must in the eyes of that alien world always be heroes, and they
must present a united front. Extending the same principles from the
material to the moral sphere, the foreigners must maintain the
reputation of their caste for probity, liberality, and trustworthiness.
Their word must be as good as their bond; they must on no account
demean themselves before the heathen, nor tolerate any temptation

from a Chinese source to take unfair advantage of their own kind,
the Caucasian or Christian, or by whatever term we may indicate the
white man. Whatever their private differences, no white man must
permit himself to acquiesce in the disparagement of his own people
in the view of the people of the country. They must be, one and all,
above suspicion. Such were some of the considerations which were
effective in maintaining the character of Europeans in China.
Although association with a race so alien as the Chinese, with such
different moral standards, must have had the usual deteriorating
effects of such contact, yet the positive gain in the formation of
character from the practice of such maxims of conduct as those
above indicated probably left a balance of advantage with the China
merchants.
The case would be imperfectly stated were mention not made of the
process of natural selection which constituted the merchants a body
of picked men. China was a remote country. It offered neither the
facility of access nor the scope for adventure which in more recent
times have attracted such streams of emigration to distant parts of
the world. The mercantile body was a close corporation,
automatically protected by barriers very difficult to surmount. The
voyage itself occupied six months. Letters were rarely answered
within a year. Hence all the machinery of business had to be
arranged with a large prescience. Even after the opening of the
overland route to Suez communication with China was maintained by
sailing-ships up till 1845, when the Lady Mary Wood, the first
steamer of the P. and O. Company, reached Hongkong, with no
accommodation for more than a few passengers, and carrying no
more cargo than a good-sized lighter. And later still, when steamers
carried the mails fortnightly to China, the expense of the trip was so
great that only a chosen few could afford it. It took £150 to £170 to
land a single man in Hongkong, and in those days when extensive
outfits were thought necessary, probably as much more had to be
laid out in that way. The merchants who established themselves in
China after the opening of the trade were either themselves men of
large means, or they were the confidential representatives of English

and American houses of great position. There were no local banks,
operations extended over one or two years, an immense outlay of
capital was required, and credit had to be maintained at an
exceedingly high level, not only as between the merchants in China
and their correspondents in London, Liverpool, New York, and
Boston, but between both and the financial centre of the world.
Through such a winnowing-machine only good grain could pass. It
was a natural result that the English and American merchants both
in China and India should have been superior as a class to the
average of other commercial communities. And what was true of
partners and heads of houses was no less so of their assistants.
There were no clerks, as the term is commonly used in England,
except Portuguese hailing from the neighbouring settlement of
Macao. The young men sent from England were selected with as
much care as it was possible to bestow, for they were precious. Not
only were they costly, but it might take a year to make good
casualties. Besides, in countries situated as China was then, where
contingencies of health were never out of mind, it was not worth
while to send out one who was a clerk and nothing more. There
must be potential capacity as well, since it could never be foreseen
how soon emergencies might arise which would require him to
assume the most responsible duties. Hence every new hand
engaged must enjoy the fullest confidence both of his immediate
employers and of the home firm to which they were affiliated.
As might be expected under such circumstances, family connections
played a large part in the selection, and the tendency of the whole
system was to minimise the gulf which in advanced societies
separates the master from the man. In education and culture they
were equals, as a consequence of which the reins of discipline might
be held lightly, all service being willingly and intelligently rendered.
The system of devolution was so fully developed that the assistant
was practically master in his own department, for the success of
which he was as zealous as the head. The mess régime under
which in most houses the whole staff, employers and employees, sat

at one table, tended strongly in the direction of a common social
level.
What still further contributed much to raise the position of assistants
was the tradition which the merchants both in India and China
inherited from the East India Company of what may be called
pampering their employees. They were permitted to carry on trade
on their own account, in the same commodities and with the same
buyers and sellers, in which they possessed advantages over their
employers in having all the firm's information at command with the
privilege of using its machinery free of cost. The abuses to which
such a system was liable are too obvious to be dwelt upon; but to be
himself a merchant, sometimes more successful than his principal,
though without his responsibilities, certainly did not detract from the
social status of the assistant.
Sixty years ago the China community was composed of men in the
prime of life. The average age was probably not over thirty—a man
of forty was a grey-beard. In this respect an evolutionary change has
come over the scene, and the average age of the adult residents
must have risen by at least ten years. But the China community in all
its stages of development has maintained the colonial characteristic
of buoyancy and hopefulness. Reverses of fortune never appalled its
members. Having been early accustomed to the alternations of fat
years and lean, a disastrous season was to them but the presage of
a bountiful one to follow; while a succession of bad years made the
reaction only the more certain. This wellspring of hope has often
helped the China merchants to carry the freshness of spring even
into the snows of winter. The nature of their pursuits, moreover,
fostered a comprehensive spirit. Trained in the school of wholesale
dealing, and habituated to work on large curves, the China
merchants have all through felt the blood of the merchant princes in
their veins, and it has even been alleged to their disadvantage that,
like the scions of decayed families the world over, the pomp and
circumstance were maintained after the material basis had in the
natural course of affairs vanished. Nay, more, that the grandiose

ideas appropriate to the heirs of a protected system have
disqualified them for the contest in small things which the latter days
have brought upon them.
Of that restricted, protected, quasi-aristocratic, half-socialistic society
some of the traditions and spirit remain; but the structure itself
could not possibly withstand the aggression of modern progress, and
it has been swept away. New elements have entered into the
composition of the mercantile and general society of the Far East, its
basis has been widened and its relations with the great world
multiplied. In innumerable ways there has been improvement, not
the least being the development of family life and the more enduring
attachment to the soil which is the result of prolonged residence.
Living, if less luxurious, is vastly more comfortable, more refined,
and more civilised, and men and women without serious sacrifices
make their home in a country which in the earlier days was but a
scene of temporary exile. Charities abound which were not before
needed; the channels of humanity have broadened, though it cannot
be said at the cost of depth, for whatever else may have changed,
the generosity of the foreign communities remains as princely as in
the good old days.
Yet is it permissible to regret some of the robuster virtues of the
generation that is past. The European solidarity vis-à-vis the Chinese
world, which continued practically unbroken into the eighth decade
of the century, a tower of moral strength to foreigners and an object
of respect to the Chinese, has now been thrown down. Not only in
private adventures have foreigners in their heat of competition let
themselves down to the level of Chinese tactics, but great financial
syndicates have immersed themselves in intrigues which either did
not tempt the men of the previous generation or tempted them in
vain; and even the Great Powers themselves have descended into
the inglorious arena, where decency is discarded like the superfluous
garments of the gladiator, and where falsity, ultra-Chinese in quality,
masquerades in Christian garb. The moral ascendancy of
Christendom has been in a hundred ways shamelessly prostituted,

leaving little visible distinction between the West and the East but
superior energy and military force.
Take them for all in all, the China merchants have been in their day
and generation no unworthy representatives of their country's
interests and policy, its manhood and character. Their patriotism has
not been toned down but expanded and rationalised by
cosmopolitan associations, and by contact with a type of national life
differing diametrically from their own. Breadth and moderation have
resulted from these conditions, and a habit of tempering the
exigencies of the day by the larger consideration of international
problems has been characteristic of the mercantile bodies in China
from first to last. And though statesmanship lies outside the range of
busy men of commerce, it must be said in justice to the merchants
of China that they have been consistently loyal to an ideal policy,
higher in its aims and more practical in its operation than that which
any line of Western statesmen, save those of Russia, has been able
to follow. It had been better if the continuous prognostications of
such a compact body of opinion had been more heeded.
II. CHINESE.
Business aptitude—High standard of commercial ethics—Circumstances hindering
great accumulations.
As it requires two to make a bargain, it would be an imperfect
account of the China trade which omitted such an important element
as the efficiency of the native trader. To him is due the fact that the
foreign commerce of his country, when uninterfered with by the
officials of his Government, has been made so easy for the various
parties concerned in it. Of all the accomplishments the Chinese
nation has acquired during the long millenniums of its history, there
is none in which it has attained to such perfect mastery as in the
science of buying and selling. The Chinese possess the Jews' passion

for exchange. All classes, from the peasant to the prince, think in
money, and the instinct of appraisement supplies to them the place
of a ready reckoner, continuously converting objects and
opportunities into cash. Thus surveying mankind and all its
achievements with the eye of an auctioneer, invisible note-book in
hand, external impressions translate themselves automatically into
the language of the market-place, so that it comes as natural to the
Chinaman as to the modern American, or to any other commercial
people, to reduce all forms of appreciation to the common measure
of the dollar. A people imbued with such habits of mind are traders
by intuition. If they have much to learn from foreigners, they have
also much to teach them; and the fact that at no spot within the
vast empire of China would one fail to find ready-made and eager
men of business is a happy augury for the extended intercourse
which may be developed in the future, while at the same time it
affords the clearest indication of the true avenue to sympathetic
relations with the Chinese. In every detail of handling and moving
commodities, from the moment they leave the hands of the
producer in his garden-patch to the time when they reach the
ultimate consumer perhaps a thousand miles away, the Chinese
trader is an expert. Times and seasons have been elaborately
mapped out, the clue laid unerringly through labyrinthine currencies,
weights, and measures which to the stranger seem a hopeless
tangle, and elaborate trade customs evolved appropriate to the
requirements of a myriad-sided commerce, until the simplest
operation has been invested with a kind of ritual observance, the
effect of the whole being to cause the complex wheels to run both
swiftly and smoothly.
To crown all, there is to be noted, as the highest condition of
successful trade, the evolution of commercial probity, which, though
no monopoly of the Chinese merchants, is one of their distinguishing
characteristics. It is that element which, in the generations before
the treaties, enabled so large a commerce to be carried on with
foreigners without anxiety, without friction, and almost without

precaution. It has also led to the happiest personal relations
between foreigners and the native trader.
When the business of the season was over [says Mr Hunter][32]
contracts were made with the Hong merchants for the next season.
They consisted of teas of certain qualities and kinds, sometimes at
fixed prices, sometimes at the prices which should be current at the
time of the arrival of the teas. No other record of these contracts was
ever made than by each party booking them, no written agreements
were drawn up, nothing was sealed or attested. A wilful breach of
contract never took place, and as regards quality and quantity the
Hong merchants fulfilled their part with scrupulous honesty and care.
The Chinese merchant, moreover, has been always noted for what
he himself graphically calls his large-heartedness, which is
exemplified by liberality in all his dealings, tenacity as to all that is
material with comparative disregard of trifles, never letting a
transaction fall through on account of punctilio, yielding to the
prejudices of others wherever it can be done without substantial
disadvantage, a sweet reasonableness, if the phrase may be
borrowed for such a purpose, which obviates disputation, and the
manliness which does not repine at the consequences of an
unfortunate contract. Judicial procedure being an abomination to
respectable Chinese, their security in commercial dealings is based
as much upon reason, good faith, and non-repudiation as that of the
Western nations is upon verbal finesse in the construction of
covenants.
Two systems so diametrically opposed can hardly admit of real
amalgamation without sacrifice of the saving principle of both. And
if, in the period immediately succeeding the retirement of the East
India Company, perfect harmony prevailed between the Chinese and
the foreign merchant, the result was apparently attained by the
foreigners practically falling in with the principles and the commercial
ethics of the Chinese, to which nothing has yet been found superior.
The Chinese aptitude for business, indeed, exerted a peculiar
influence over their foreign colleagues. The efficiency and alacrity of
the native merchants and their staff were such that the foreigners

fell into the way of leaving to them the principal share in managing
the details of the business. When the venerable, but unnatural, Co-
hong system of Old Canton was superseded by the compradoric, the
connection between the foreign firm and their native staff became
so intimate that it was scarcely possible to distinguish between the
two, and misunderstandings have not unfrequently arisen through
third parties mistaking the principal for the agent and the agent for
the principal.
Such a relationship could not but foster in some cases a certain
lordly abstraction on the part of the foreign merchant, to which
climatic conditions powerfully contributed. The factotum, in short,
became a minister of luxury, everywhere a demoralising influence,
and thus there was a constant tendency for the Chinese to gain the
upper hand,—to be the master in effect though the servant in name.
The comprador was always consulted, and if the employer ventured
to omit this formality the resulting transaction would almost certainly
come to grief through inexplicable causes. Seldom, however, was his
advice rejected, while many of the largest operations were of his
initiation. Unlimited confidence was the rule on both sides, which
often took the concrete form of considerable indebtedness, now on
the one side now on the other, and was regularly shown in the
despatch of large amounts of specie into the far interior of the
country for the purchase of tea and silk in the districts of their
growth. For many years the old practice was followed of contracting
for produce as soon as marketable, and sometimes even before.
During three or four months, in the case of tea, large funds
belonging to foreign merchants were in the hands of native agents
far beyond the reach of the owners, who could exercise no sort of
supervision over the proceedings of their agents. The funds were in
every case safely returned in the form of produce purchased, which
was entered to the foreign merchant at a price arbitrarily fixed by
the comprador to cover all expenses. Under such a régime it would
have needed no great perspicacity, one would imagine, to foretell in
which pocket the profits of trading would eventually lodge. As a
matter of fact, the comprador generally grew rich at the expense of

his employer. All the while the sincerest friendship existed between
them, often descending to the second or third generation.[33]
It would be natural to suppose that in such an extensive commercial
field as the empire of China, exploited by such competent traders,
large accumulations of wealth would be the result. Yet after making
due allowance for inducements to concealment, the wealth even of
the richest families probably falls far short of that which is not
uncommon in Western countries. Several reasons might be adduced
for the limitation, chiefly the family system, which necessitates
constant redistribution, and which subjects every successful man to
the attentions of a swarm of parasites, who, besides devouring his
substance with riotous living, have the further opportunity of ruining
his enterprises by their malfeasance. Yet although individual wealth
may, from these and other causes, be confined within very moderate
limits, the control of capital for legitimate business is ample. Owing
to the co-operative system under which the financiers of the country
support and guarantee each other, credit stands very high, enabling
the widely ramified commerce of the empire to be carried on upon a
very small nucleus of cash capital. The banking organisation of China
is wonderfully complete, bills of exchange being currently negotiable
between the most distant points of the empire, the circulation of
merchandise maintaining the equilibrium with comparatively little
assistance from the precious metals.
The true characteristics of a people probably stand out in a clearer
light when they are segregated from the conventionalities of their
home and forced to accommodate themselves to unaccustomed
conditions. Following the Chinese to the various commercial colonies
which they have done so much to develop, it will be found that they
have carried with them into their voluntary exile the best elements
of their commercial success in their mother country. The great
emporium of Maimaichên, on the Siberian frontier near Kiachta, is an
old commercial settlement mostly composed of natives of the
province of Shansi, occupying positions of the highest respect both
financially and socially. The streets of the town are regular, wide,

and moderately clean. The houses are solid, tidy, and tasteful, with
pretty little courtyards, ornamental door-screens, and so forth, the
style of the whole being described as superior to what is seen in the
large cities within China proper. The very conditions of exile seem
favourable to a higher scale of living, free alike from the incubus of
thriftless relations and from the malign espionage of Government
officials.
In the Philippine Islands and in Java the Chinese emigrants from the
southern provinces have been the life and soul of the trade and
industry of these places. So also in the British dominions, as at
Singapore and Penang, which are practically Chinese Colonies under
the British flag. Hongkong and the Burmese ports are of course no
exceptions.
The description given by Mr Thomson[34] of the Chinese in Penang
would apply equally to every part of the world in which the Chinese
have been permitted to settle:—
Should you, my reader, ever settle in Penang, you will be there
introduced to a Chinese contractor who will sign a document to do
anything. His costume will tell you that he is a man of inexpensive yet
cleanly habits. He will build you a house after any design you choose,
and within so many days, subject to a fine should he exceed the
stipulated time. He will furnish you with a minute specification, in
which everything, to the last nail, will be included. He has a brother
who will contract to make every article of furniture you require, either
from drawings or from models. He has another brother who will fit
you and your good lady with all sorts of clothing, and yet a third
relative who will find servants, and contract to supply you with all the
native and European delicacies in the market upon condition that his
monthly bills are regularly honoured.
It is, indeed, to Chinamen that the foreign resident is indebted for
almost all his comforts, and for the profusion of luxuries which
surround his wonderfully European-looking home on this distant
island.
The Chinese are everywhere found enterprising and trustworthy men
of business. Europeans, worried by the exhaustless refinements of

the Marwarree or Bengali, find business with the Chinese in the
Straits Settlements a positive luxury. Nor have the persecutions of
the race in the United States and in self-governing British colonies
wholly extinguished the spark of honour which the Chinese carry
with them into distant lands. An old 'Forty-niner, since deceased,
related to the writer some striking experiences of his own during a
long commercial career in San Francisco. A Chinese with whom he
had dealings disappeared from the scene, leaving a debt to Mr
Forbes of several thousand dollars. The account became an eyesore
in the books, and the amount was formally written off and
forgotten. Some years after, Mr Forbes was surprised by a visit from
a weather-beaten Chinese, who revealed himself as the delinquent
Ah Sin and asked for his account. Demurring to the trouble of
exhuming old ledgers, Mr Forbes asked Ah Sin incredulously if he
was going to pay. Why, certainly, said the debtor. The account was
thereupon rendered to him with interest, and after a careful
examination and making some corrections, Ah Sin undid his belt and
tabled the money to the last cent, thereupon vanishing into space
whence he had come.

CHAPTER XIV.
HONGKONG.
Two British landmarks—Chinese customs and Hongkong—Choice of the island—
Vitality of colony—Asylum for malefactors—Chinese official hostility—
Commanding commercial position—Crown Colony government—Management of
Chinese population—Their improvement—English education—Material progress—
Industrial institutions—Accession of territory.
The past sixty years of war and peace in China have left two
landmarks as concrete embodiments of British policy—the Chinese
maritime customs and the colony of Hongkong. These are
documents which testify in indelible characters both to the motives
and to the methods of British expansion throughout the world. For
good and for evil their record cannot be explained away. Both
institutions are typically English, inasmuch as they are not the
fulfilment of a dream or the working out of preconcerted schemes,
but growths spontaneously generated out of the local conditions,
much like that of the British empire itself, and with scarcely more
conscious foresight on the part of those who helped to rear the
edifice.
The relation of the British empire to the world, which defies
definition, is only revealed in scattered object-lessons. India throws
some light upon it—the colonies much more; and though in some
respects unique in its character, Hongkong in its degree stands
before the world as a realisation of the British ideal, with its faults
and blunders as well as with its excellences and successes.

The want of a British station on the China coast had long been felt,
and during the ten years which preceded the cession innumerable
proposals were thrown out, some of which distinctly indicated
Hongkong itself as supplying the desideratum. But as to the status
of the new port the various suggestions made neutralised each
other, until the course of events removed the question out of the
region of discussion and placed it in the lap of destiny.
The earliest English visitors to the island described it as inhabited by
a few weather-beaten fishermen, who were seen spreading their
nets and drying their catch on the rocks. Cultivation was restricted to
small patches of rice, sweet-potatoes, and buckwheat. The
abundance of fern gave it in places an appearance of verdure, but it
was on the whole a treeless, rugged, barren block of granite. The
gentlemen of Lord Amherst's suite in 1816, who have left this
record, made another significant observation. The precipitous island,
twelve miles long, with its deep-water inlets, formed one side of a
land-locked harbour, which they called Hongkong Sound, capable of
sheltering any number of ships of the largest size. Into this
commodious haven the English fugitives, driven first from Canton
and then from Macao, by the drastic decree of the Chinese
authorities in 1839, found a refuge for their ships, and afterwards a
footing on shore for themselves. Stern necessity and not their wills
sent them thither. The same necessity ordained that the little band,
once lodged there, should take root, and growth followed as the
natural result of the inherent vitality of the organism. As Dr Eitel well
points out, this small social body did not originate in Hongkong: it
had had a long preparatory history in Macao, and in the Canton
factories, and may be considered, therefore, in the light of a healthy
swarm from the older hives.
During the first few years of the occupation the selection of the
station was the subject of a good deal of cheap criticism in the
press. A commercial disappointment and a political failure, it was
suggested by some that the place should be abandoned. It was
contrasted unfavourably with the island of Chusan, which had been

receded to China under the same treaty which had ceded Hongkong
to Great Britain; and even as late as 1858 Lord Elgin exclaimed,
How anybody in their senses could have preferred Hongkong to
Chusan seems incredible.
But, in point of fact, there had been little or no conscious choice in
the matter. The position may be said to have chosen itself, since no
alternative was left to the first British settlers. As for Chusan, it had
been occupied and abandoned several times. The East India
Company had an establishment there in the beginning of the
eighteenth century, and if that station was finally given up either on
its merits or in favour of Hongkong, it was certainly not without
experience of the value of the more northerly position. Whatever
hypothetical advantages, commercial or otherwise, might have
accrued from the retention of Chusan, the actual position attained by
Hongkong as an emporium of trade, a centre of industry, and one of
the great shipping ports in the world, furnishes an unanswerable
defence both of the choice of the site and the political structure
which has been erected on it. Canton being at once the centre of
foreign trade and the focus of Chinese hostility, vicinity to that city
was an indispensable condition of the location of the British
entrepot, and the place of arms from which commerce could be
defended. And it would be hard even now to point to any spot on
the Chinese coast which fulfilled the conditions so well as Hongkong.
The course of its development did not run smooth. It was not to be
expected. The experiment of planting a British station in contact with
the most energetic as well as the most turbulent section of the
population of China was not likely to be carried out without
mistakes, and many have been committed. Indeed, from the day of
its birth down to the present time domestic dissensions and
recriminations respecting the management of its affairs have never
ceased.
This was inevitable in a political microcosm having neither diversity
of interest nor atmospheric space to soften the perspective. The
entire interests of the colony were comprised within the focal

distance of myopic vision. Molehills thus became mountains, and the
mote in each brother's eye assumed the dimensions of animalcula
seen through a microscope. The bitter feuds between the heads of
the several departments of the lilliputian Government which
prevailed during the first twenty years must have been fatal to any
young colony if its progress had depended on the wisdom of its
rulers. Happily a higher law governs all these things.
Freedom carried with it the necessary consequences, and for many
years the new colony was a tempting Alsatia for Chinese
malefactors, an asylum for pirates, who put on and off that character
with wonderful facility, and could hatch their plots there fearless of
surveillance. When the Taiping rebellion was at its height, piracy
became so mixed with insurrection that the two were not
distinguishable, and it required both firmness and vigilance on the
part of the authorities to prevent the harbour of Hongkong becoming
the scene of naval engagements between the belligerents. During
the hostilities of 1857-58 a species of dacoity was practised with
impunity by Chinese, who were tempted by rewards for the heads of
Englishmen offered by the authorities of Canton.
It cannot, therefore, be denied that the immigrants from the
mainland in the first and even the second decade of its existence
were leavened with an undesirable element, causing anxiety to the
responsible rulers.
The Chinese authorities, as was natural, waged relentless war on the
colony from its birth. Though compelled formally to admit that the
island and its dependencies were a British possession, they still
maintained a secret authority over the Chinese who settled there,
and even attempted to levy taxes. As they could not lay hands on its
trade, except the valuable portion of it which was carried on by
native craft, they left no stone unturned to destroy that. By skilful
diplomacy, for which they are entitled to the highest credit, they
obtained control over the merchant junks trading to Hongkong, and
imposed restrictions on them calculated to render their traffic
impossible. By the same treaty they obtained the appointment of a

British officer as Chinese revenue agent in Hongkong—a concession,
however, disallowed by the good sense of the British Government.
But the Chinese were very tenacious of the idea of making
Hongkong a customs station, never relaxing their efforts for forty
years, until the convention of 1886 at last rewarded their
perseverance by a partial fulfilment of their hopes.
For reasons which, if not very lofty, are yet very human, the
diplomatic and consular agents of Great Britain have never looked
sympathetically on the colony—indeed have often sided with the
Chinese in their attempts to curtail its rights.
Nor has the Home Government itself always treated the small colony
with parental consideration. Before it was out of swaddling-clothes
the Treasury ogre began to open his mouth and, like the East India
Company, demand remittances. A military establishment was
maintained on the island, not for the benefit of the residents, but for
the security of a strategical position in the imperial system. The
colonists were mulcted in a substantial share of the cost, which the
governor was instructed to wring out of them. The defences
themselves, however, were neglected, and allowed to grow obsolete
and useless, and, if we mistake not, it was the civil community, and
not the Government, that insisted on their being modernised. The
compromise eventually arrived at was, that the colonists provided
the guns and the imperial Government the forts. An interesting
parallel to this was the case of Gibraltar, which possessed no dock
until the civil community by sheer persistence, extending over many
years, at length overcame the reluctance of the British Government
to provide so essential an adjunct to its naval establishment. The
colony had suffered much from the war with China, but the Home
Government refused it any participation in the indemnity extorted
from the Chinese.
But these and other drawbacks were counterbalanced, and
eventually remedied, by the advantages offered by a free port and a
safe harbour. Standing in the fair way of all Eastern commerce,
which pays willing tribute to the colony, Hongkong attracted trade

from all quarters in a steadily increasing volume, and became the
pivot for the whole ocean traffic of the Far East.[35] The tide of
prosperity could not be stayed—it invaded every section of the
community. The character of the Chinese population was
continuously raised. The best of them accumulated wealth: the
poorest found remunerative employment for their labour. Crime, with
which the colony had been tainted, diminished as much through the
expulsive power of material prosperity as from the judicious
measures of the executive Government, for the credit must not be
denied to successive administrators for the improvement in the
condition of the colony. Among those none was more deserving of
praise than Sir Richard MacDonnell (1865-72), who on catching
sight, as he entered the harbour, of an enormous building, which he
was told was the jail, remarked, I will not fill that, but stop the
crime; and he was nearly as good as his word,—a terror to evil-
doers.
A Crown colony is the form of government which challenges the
most pungent criticism. The elected members of its legislature, being
a minority, can only in the last resort acquiesce in the decisions of
the official majority who constitute the executive Government. Such
a minority, however, is by no means wanting in influence, for it is,
after all, publicity which is the safeguard of popular liberty. The
freedom of speech enjoyed by an Opposition which has no fear of
the responsibility of office before its eyes widens the scope of its
criticisms, and imparts a refreshing vigour to the invective of those
of its members who possess the courage of their convictions. It
reaches the popular ear, and the apprehension of an adverse public
opinion so stimulated can never fail to have its effect on the acts of
the Administration. Under such a régime it seems natural that, other
things being equal, each governor in turn should be esteemed the
worst who has borne rule in the colony, and in any case his merits
are never likely to be fairly gauged by any local contemporary
estimate. King Stork, though fair and far-seeing, may be more
obnoxious to criticism than King Log, who makes things pleasant
during his official term.

Hongkong being established as a free port, the functions of
Government were practically limited to internal administration, and
the question of greatest importance was the control of the Chinese
population which poured in. This was a new problem. Chinese
communities had, indeed, settled under foreign rule before, as in the
Straits Settlements, in Java, and in Manila, but at such distances
from their home as rendered the settlers amenable to any local
regulations which might be imposed on them. Distance even acted
as a strainer, keeping back the dregs. But Hongkong was nearer to
China than the Isle of Wight is to Hampshire. Evil-doers could come
and go at will. It could be overrun in the night and evacuated in the
morning. Spies were as uncontrollable as house-flies, and whenever
it suited the Chinese Government to be hostile, they proved their
power to establish such a reign of terror in the colony that it was
dangerous to stray beyond the beat of the armed policeman. Clearly
it was of primary importance to come to terms with the native
community, to reduce them to discipline, to encourage the good and
discourage the bad among the Chinese settlers. As their numbers
increased the public health demanded a yet stricter supervision of
their habits. Sanitary science had scarcely dawned when the colony
was founded, and its teachings had to be applied, as they came to
light, to conditions of life which had been allowed to grow up in
independence of its requirements. To tolerate native customs,
domestic habits, and manner of living, while providing for the
general wellbeing of a community in a climate which at its best is
debilitating, taxed the resources of the British executive, and of
course gave rise to perpetual recrimination. But the thing has been
accomplished. Successive conflagrations have co-operated with the
march of sanitary reform and the advance in their worldly
circumstances in so improving the dwellings of the population, that
their housing now compares not unfavourably with that of the native
cities of India. The Southern Chinese are naturally cleanly, and
appreciative of good order when it is judiciously introduced among
them, even from a foreign source.

A more complex question was that of bringing an alien population
such as the Chinese within the moral pale of English law, for law is
vain unless it appeals to the public conscience. The imposition of
foreign statutes on a race nursed on oral tradition and restrained
from misdoing by bonds invisible to their masters was not an
undertaking for which success could be safely foretold. The effect of
a similar proceeding on the subtle natives of India has been
described as substituting for a recognised morality a mere game of
skill, at which the natives can give us long odds and beat us. The
mercantile and money-lending classes in India, says Mr S. S.
Thorburn, delight in the intricacy and surprises of a good case in
court. With the Chinese it has been otherwise. The population of
Hongkong have so far assimilated the foreign law that, whether or
not it satisfies their innate sense of right, it at least governs their
external conduct, and crime has been reduced very low: as for
litigation, it is comparatively rare; it is disreputable, and has no place
in the Chinese commercial economy.
The best proofs of their acceptance of colonial rule is the constantly
increasing numbers of the Chinese residents; the concentration of
their trading capital there; their investments in real estate and in
local industries; their identification with the general interests of the
colony, and their adopting it as a home instead of a place of
temporary exile. The means employed to conciliate the Chinese must
be deemed on the whole to have been successful. There was first
police supervision, then official protection under a succession of
qualified officers, then representation in the Colonial Legislature and
on the commission of the peace. The colonial executive has wisely
left to the Chinese a large measure of a kind of self-government
which is far more effective than anything that could find its
expression in votes of the Legislature. The administration of purely
Chinese affairs by native committees, with a firm ruling hand over
their proceedings, seems to fulfil every purpose of government. The
aim has been throughout to ascertain and to gratify, when
practicable, the reasonable wants of the Chinese, who have
responded to these advances by an exhibition of public spirit which

no society could excel. It is doubtful whether in the wide dominion
of the Queen there are 250,000 souls more appreciative of orderly
government than the denizens of the whilom nest of pirates and cut-
throats—Hongkong.
As an educational centre Hongkong fulfils a function whose value is
difficult to estimate. From the foundation of the colony the subject
engaged the attention of the executive Government, as well as of
different sections of the civil community. The missionary bodies were
naturally very early in the field, and there was for a good many
years frank co-operation between them and the mercantile
community in promoting schools both for natives and Europeans. In
time, however, either their aims were found to diverge or else their
estimate of achievement differed, and many of the missionary
teaching establishments were left without support.
After an interval of languor, however, new life was infused into the
educational schemes of the colony. The emulation of religious sects
and the common desire to bring the lambs of the flock into their
respective folds inspired the efforts of the propagandists, their zeal
reacting on the colonial Government itself with the most gratifying
results, so far at least as the extension of the field of their common
efforts was concerned.
The Chinese had imported their own school systems, while taking
full advantage of the educational facilities provided by the
Government and the Christian bodies. Being an intellectual race,
they are well able to assimilate the best that Christendom has to
offer them. But the colonial system contents itself with a sound
practical commercial education, which has equipped vast numbers of
Chinese for the work of clerks, interpreters, and so forth, and has
thus been the means of spreading the knowledge of the English
language over the coast of China, and of providing a medium of
communication between the native and European mind.
The material progress of Hongkong speaks volumes for the energy
of its community. The precipitous character of the island left scarcely

a foothold for business or residential settlement. The strip which
formed the strand front of the city of Victoria afforded room for but
one street, forcing extensions up the rugged face of the hill which
soon was laid out in zig-zag terraces: foundations for the houses are
scarped out of the rock, giving them the appearance of citadels. The
locality being subject to torrential rains, streets and roads had to be
made with a finished solidity which is perhaps unmatched. Bridges,
culverts, and gutters all being constructed of hewn granite and fitted
with impervious cement, the storm-waters are carried off as clean as
from a ship's deck. These municipal works were not achieved
without great expense and skilfully directed labour, of which an
unlimited supply can always be depended on. And the credit of their
achievement must be equally divided between the Government and
the civil community.
The island is badly situated as regards its water-supply, which has
necessitated the excavation of immense reservoirs on the side
farthest from the town, the aqueduct being tunnelled for over a mile
through a solid granite mass. These and other engineering works
have rendered Hongkong the envy of the older colonies in the Far
East. No less so the palatial architecture in which the one natural
product of the island has been turned to the most effective account.
The quarrying of granite blocks, in which the Chinese are as great
adepts as they are in dressing the stones for building, has been so
extensive as visibly to alter the profile of the island.
A great deficiency of the island as a commercial site being the
absence of level ground, the enterprise of the colonists has been
incessantly directed towards supplying the want. Successive
reclamations on the sea-front, costing of course large sums of
money, have so enlarged the building area that the great
thoroughfare called Queen's Road now runs along the back instead
of the front of a new city, the finest buildings of all being the most
recent, standing upon the newly reclaimed land. It is characteristic
of such improvements, that, while in course of execution, they
should be deemed senseless extravagance, due to the ambition of

some speculator or the caprice of some idealist, thus perpetually
illustrating the truth of the Scottish saying, Fules and bairns should
never see a thing half done. Hongkong has been no exception to so
universal a rule.
The industrial enterprise of the colony has fully kept pace with its
progress in other respects. The Chinese quarter resembles nothing
so much as a colony of busy ants, where every kind of handicraft is
plied with such diligence, day in and day out, as the Chinese alone
seem capable of. The more imposing works conducted by foreigners
occupy a prominent place in the whole economy of the Far East.
Engineering and shipbuilding have always been carried on in the
colony. Graving-docks capable of accommodating modern
battleships, and of executing any repairs or renewals required by
them as efficiently as could be done in any part of the world,
constitute Hongkong a rendezvous for the navies of all nations.
Manufactures of various kinds flourish on the island. Besides cotton-
mills, some of the largest sugar-refineries in the world, fitted with
the most modern improvements, work up the raw material from
Southern China, Formosa, the Philippines, and other sugar-growing
countries in the Eastern Archipelago, thus furnishing a substantial
item of export to the Australian colonies and other parts of the
world. The colony has thereby created for itself a commerce of its
own, while its strategical situation has enabled it to retain the
character of a pivot on which all Far Eastern commerce turns.
This pivotal position alone, and not the local resources of the place,
enabled the colony to found one of the most successful financial
organisations of the modern world. The Hongkong and Shanghai
Bank has had a history not dissimilar from that of the colony as a
whole, one of success followed by periods of alternate depression
and elation. Now in the trough of the wave and now on its crest, the
bank has worked its way by inherent vitality through all vicissitudes
of good or bad fortune, until it has gone near to monopolising the
exchange business of the Far East, and has become the recognised

medium between the money-market of London and the financial
needs of the Imperial Chinese administration.
It should not be overlooked as a condition of its success that the
great Hongkong Bank, like all other successful joint-stock
enterprises, whether in Hongkong or in China, has from its origin
borne a broad international character. Though legally domiciled in a
British possession, representative men of all nationalities sit on its
board and take their turn in the chairmanship as it comes round.
The international character, indeed, may be cited as one of the
elements of the success of the colony itself. No disability of any kind
attaches to alien settlers, not even exclusion from the jury panel.
They are free to acquire property, to carry on business, to indulge
their whims, and to avail themselves of all the resources of the
colony, and enjoy the full protection of person and property which
natural-born British subjects possess. They come and go at their
pleasure, no questions asked, no luggage examined, no permits
required for any purpose whatever coming within the scope of
ordinary life. Nor are they even asked whether they appreciate these
advantages or not; in fact they are as free to criticise the institutions
under which they live as if they had borne their part in creating
them, which, in fact, they have done, and this it is which marks the
vitality of the British system, whether in the mother country or in its
distant dependencies.
The exceedingly cramped conditions of life on the island having
proved such an obstacle to its development, the acquisition of a
portion of the mainland forming one side of the harbour was at an
early period spoken of as a desideratum for the colony. The idea
took no practical shape, however, until the occupation of Canton by
the Allied forces under the administration of Consul Parkes; and it is
one of the most noteworthy achievements of that indefatigable man
that, during the time when Great Britain was in fact at war with the
Government of China, he should have succeeded, on his own
initiative, in obtaining from the governor of the city a lease of a
portion of land at Kowloon, which was subsequently confirmed by

the convention of Peking in 1860. The improvement of artillery and
other means of attack on sea-forts left the island very vulnerable,
and the measures taken by the various European Powers to establish
naval stations on the Chinese coast, together with the efforts which
the country itself was making to become a modern military Power,
rendered it a matter of absolute necessity, for the preservation of
the island, that a sufficient area of the adjacent territory should be
included within its defences. Following the example set by Germany
and Russia, the British Government concluded an arrangement with
the Government of China by which the needed extension was
secured to Great Britain under a ninety-nine years lease. A
convention embodying this agreement was signed at Peking in June
1898.

CHAPTER XV.
MACAO.
Contrast with Hongkong—An interesting survival—Trading facilities—Relations with
Chinese Government—Creditable to both parties—Successful resistance to the
Dutch—Portuguese expulsion from Japan—English trading competitors enjoy
hospitality of Macao—Trade with Canton—Hongkong becomes a rival—Macao
eclipsed—Gambling, Coolie trade, Piracy—Population—Cradle of many
improvements—Distinguished names.
The three hours' transit from Hongkong to Macao carries one into
another world. The incessant scream of steam-launches which
plough the harbour in all directions night and day gives place to the
drowsy chime of church bells, and instead of the throng of busy
men, one meets a solitary black mantilla walking demurely in the
middle of a crooked and silent street. Perhaps nowhere is the
modern world with its clamour thrown into such immediate contrast
with that which belongs to the past.
The settlement of Macao is a monument of Chinese toleration and of
Portuguese tenacity. The Portuguese learnt at an early stage of their
intercourse the use of the master-key to good relations with the
Chinese authorities. It was to minister freely to their cupidity, which
the Portuguese could well afford to do out of the profits of their
trading. To maintain ourselves in this place we must spend much
with the Chinese heathen, as they themselves said in 1593 in a
letter to Philip I. Macao is, besides, an interesting relic of that heroic
age when a new heaven and a new earth became the dream of
European adventurers. The spot was excellently well suited for the
purposes, commercial and propagandist, which it was destined to

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