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The Binomial distribution looks at n trials "with replacement." The hypergeometric distribution is for the case "without replacement."

Here p changes from one Bernoulli trial to the next. Specifically, we have a population of size N with M out of the N members being "Successes" and the remaining (N-M) being "Failures." We choose a random sample of n (equivalent to taking out n members in succession without replacement).

The hypergeometric variable X refers to the number of "successes" in the sample of size n.

X can take on any value x in {Max[0,n-(N-M)] to Min(n,M)}.

The p.m.f of X depends on three parameters: n, M and N. It is denoted by b(x; n,M,N) and is the probability that X=x and given by

The expected value and variance of X are given by

When the ratio n/N is small (typically £ 0.05) the hypergeometric distribution may be approximated by the binomial distribution.

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