Bernoulli and binomial random variables are used to model success/failure experiments. A Bernoulli variable represents a single trial with outcomes success (1) and failure (0). A binomial variable counts the number of successes in n independent Bernoulli trials. The probability of x successes in n trials is given by the binomial distribution. Its mean and variance can be derived. The moment generating function of the binomial distribution helps compute moments like variance.

1. 1.10 Bernoulli’s random Variables & Binomial Distribution

2. Bernoulli Random Variable Suppose 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 Variable Suppose 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 Distribution Bernoulli Trials There 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 – x The 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 by This 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 formula since 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 Function Table for n = 2 and 3 and p = .05 to .25

12. Example

13. The Mean and the Variance of a Probability Distribution Mean of discrete probability distribution The 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 distribution p  probability of success n  number of trials Variance of binomial distribution

15. The Mean and the Variance of a Probability Distribution Mean of binomial distribution p  probability of success n  number of trials Proof:

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 variance Variance of binomial distribution Proof:

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 summation The 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 distribution Moment Generating function for Binomial distribution