Lecture_16 - 1 BERNOULLI AND BINOMIAL DISTRIBUTIONS Lecture...

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1 BERNOULLI AND BINOMIAL DISTRIBUTIONS Lecture 16 ORIE3500/5500 Summer2009 Chen Class Today Bernoulli and Binomial Distribution Hypergeometric Distribution 1 Bernoulli and Binomial distributions A Bernoulli is the simplest random variable of all(after the constant). It takes the values 1 and 0 with probabilities p and 1 - p respectively. One views this as whether an experiment is a success or a failure. It is 1 in case of success and 0 in case of a failure. So if X is a Bernoulli( p ) random variable then the pmf of X is p X (0) = 1 - p, p X (1) = p. We also saw earlier that E ( X ) = p , E ( X 2 ) = p and var ( X ) = p (1 - p ). The cdf of X will be F X ( x ) = 0 x < 0 , 1 - p 0 x < 1 , 1 1 x. The moment generating function of X is φ X ( t ) = (1 - p ) + pe t . Now suppose there are n independent experiments and each can be a success with probability p and a failure with probability 1 - p . We might be interested in the probability distribution. Let X be the total number of successful experiments. If Y i is 1 if the i th experiment is a success, then X = Y 1 + · · · + Y n ., where Y i ’s are independent Bernoulli( p ) random variables. This X is said to be a binomial random variable with parameters n and p , bin ( n, p ). A bin ( n, p ) random variable X can take the values 0 , 1 , . . . , n and
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