![Binomial Distribution Thus the probability of getting x successes in n trials is given byThis probability distribution is called the binomial distribution because for x = 0, 1, 2, …, and n the value of the probabilities are successive terms of binomial expansion of [p + (1 – p)]n;](https://image.slidesharecdn.com/1-10bernoullisrandomvariablesandbinomialdistribution-100217112325-phpapp02/85/Bernoullis-Random-Variables-And-Binomial-Distribution-7-320.jpg)
Bernoullis Random Variables And Binomial Distribution
- 1. 1.10 Bernoulli’s random Variables & Binomial Distribution
- 2. Bernoulli Random VariableSuppose that a trial, or an experiment, whose outcome can be classified as either a success or a failure is performed. If we let X=1 when the outcome is a success and X=0 when the outcome is a failure, then the pmf of X is given by
- 3. Bernoulli Random Variable A random variable X is said to be a Bernoulli random variable (after the Swiss mathematician James Bernoulli) if its probability mass function is given by
- 4. Binomial Random VariableSuppose now that n independent trials, each of which results in a success with probability p and in a failure with probability 1-p, are to be performed. If X represents the number of successes that occur in the n trials, then X is said to be a Binomial random variable with parameters (n,p) . Thus a Bernoulli random variable is just a binomial random variable with parameters (1,p) .
- 5. Binomial DistributionBernoulli TrialsThere are only two possible outcomes for each trial.The probability of a success is the same for each trial.There are n trials, where n is a constant.The n trials are independent.
- 6. Binomial Distribution Let X be the random variable that equals the number of successes in n trials.If p and 1 – p are the probabilities of success and failure on any one trial then the probability of getting x successes and n – x failures in some specific order is px(1- p)n – xThe number of ways in which one can select the x trials on which there is to be a success is
- 7. Binomial Distribution Thus the probability of getting x successes in n trials is given byThis probability distribution is called the binomial distribution because for x = 0, 1, 2, …, and n the value of the probabilities are successive terms of binomial expansion of [p + (1 – p)]n;
- 8. Binomial Distribution for the same reason, the combinatorial quantities are referred to as binomial coefficients. The preceding equation defines a family of probability distributions with each member characterized by a given value of the parameterp and the number of trials n.
- 9. Binomial Distribution Distribution function for binomial distribution
- 10. Binomial Distribution The value of b(x;n,p) can be obtained by formulasince the two cumulative probabilities B(x; n, p) and B(x - 1; n, p) differ by the single term b(x; n,p).If n is large the calculation of binomial probability can become quite tedious.
- 11. Binomial Distribution FunctionTable for n = 2 and 3 and p = .05 to .25
- 12. Example
- 13. The Mean and the Variance of a Probability DistributionMean of discrete probability distributionThe mean of a probability distribution is the mathematical expectation of a corresponding random variable. If a random variable X takes on the values x1, x2, …, or xk, with the probability f(x1), f(x2),…, and f(xk), its mathematical expectation or expected value is = x1· f(x1) + x2· f(x2) + … + xk· f(xk)
- 14. The Mean and the Variance of a Probability Distribution Mean of binomial distributionp probability of successn number of trialsVariance of binomial distribution
- 15. The Mean and the Variance of a Probability Distribution Mean of binomial distributionp probability of successn number of trialsProof:
- 16. The Mean and the Variance of a Probability Distribution Put x – 1= y and n – 1 = m, so n – x = m – y,
- 17. Computing formula for the varianceVariance of binomial distributionProof:
- 18. Put x – 1 = y and n – 1 = m The Mean and the Variance of a Probability Distribution
- 19. The Mean and the Variance of a Probability Distribution
- 20. Put y – 1 = z and m – 1 = l in first summationThe Mean and the Variance of a Probability Distribution
- 21. Moment Generating function for Binomial distribution
- 22. Second ordinary/raw moment (moment about origin)Moment Generating function for Binomial distribution
- 23. Moment Generating function for Binomial distributionMoment Generating function for Binomial distribution