![The density function of X is given by
f(x) = a + bx2 0<=x<=1
0 otherwise
If E[X] = 3/5 , find a and b](https://image.slidesharecdn.com/lecture20hinglish-250605060305-1338de28/85/expectation-and-variance-of-C-R-V-in-prob-and-stat-3-320.jpg)
![Properties of Expectation
i. Let g and h be functions, and let a and b be constants. For any random variable X
(discrete or continuous),
E {ag(X) + bh(X) } = aE { g(X) } + bE { h(X) } .
In particular, E(aX + b) = aE(X) + b.
ii. Let X and Y be ANY random variables (discrete, continuous, independent, or non-
independent). Then E(X + Y ) = E(X) + E(Y ).
iii. Let X and Y be independent random variables, and g, h be functions.
Then E(XY ) = E(X)E(Y )
E [ g(X)h(Y ) ] = E [ g(X) ] E [ h(Y ) ] .](https://image.slidesharecdn.com/lecture20hinglish-250605060305-1338de28/85/expectation-and-variance-of-C-R-V-in-prob-and-stat-5-320.jpg)
![If a random variable X is defined such that E[(X − 1)2 ] = 10 and E[(X − 2)2 ]=6,
find mean and variance](https://image.slidesharecdn.com/lecture20hinglish-250605060305-1338de28/85/expectation-and-variance-of-C-R-V-in-prob-and-stat-11-320.jpg)
expectation and variance of C.R.V in prob and stat
- 1. Expectation and Variance of Continuous Random Variable 0 1
- 3. The density function of X is given by f(x) = a + bx2 0<=x<=1 0 otherwise If E[X] = 3/5 , find a and b
- 5. Properties of Expectation i. Let g and h be functions, and let a and b be constants. For any random variable X (discrete or continuous), E {ag(X) + bh(X) } = aE { g(X) } + bE { h(X) } . In particular, E(aX + b) = aE(X) + b. ii. Let X and Y be ANY random variables (discrete, continuous, independent, or non- independent). Then E(X + Y ) = E(X) + E(Y ). iii. Let X and Y be independent random variables, and g, h be functions. Then E(XY ) = E(X)E(Y ) E [ g(X)h(Y ) ] = E [ g(X) ] E [ h(Y ) ] .
- 6. Suppose that X and Y are independent random variables with probability densities and g(x) = 8/x3 , x> 2, 0, elsewhere, and h(y) = 2y, 0 <y< 1, 0, elsewhere. Find the expected value of Z = XY .
- 7. Variance The variance of a random variable X is a measure of how spread out it is.
- 8. GATE DA Sample paper 2024 X is a uniformly distributed random variable from 0 to 1 The variance of X is (A) 1/2 (B) 1/3 (C) 1/4 (D) 1/12
- 10. Properties of Variance (i) Let g be a function, and let a and b be constants. For any random variable X (discrete or continuous), Var { ag(X) + b } = a2 Var { g(X) } . In particular, Var(aX + b) = a2 Var(X). (ii) Let X and Y be independent random variables. Then Var(X + Y ) = Var(X) + Var(Y ).
- 11. If a random variable X is defined such that E[(X − 1)2 ] = 10 and E[(X − 2)2 ]=6, find mean and variance
- 14. <=x
- 16. <=x
- 20. A random variable X has the following probability distribution. x : 0 1 2 3 4 5 6 7 p(x) 0 K 2K 2K 3K K2 2K2 7K² + K Find the smallest value of λ for which P(X≤ λ) > 1/2.
- 21. If the pdf of a RV X is given by f(x)= 1/4 in-2<x<2 0 otherwise Find P(|X|>1)
- 22. The diameter of an electric cable X is a continuous RV with pdf f(x) = kx(1- x), 0 ≤ x ≤ 1. Find (i) the value of a such that P(X < a) = 2P(X> a) (ii) P(X≤ 1/2 / 1/3 < X < 2/3).
- 23. If the weather is good (which happens with probability 0.6), Alice walks the 2 miles to class at a speed of V = 5 miles per hour, and otherwise drives her motorcycle at a speed of V = 30 miles per hour. What is the mean of the time T to get to class?
- 24. Thank you