The Hypergeometric Distribution: Miscellaneous Notes
Consider the problem of drawing a sample without replacement of size n from a population of
size N. Suppose that the population contains r items which are designated as a success and N-r
items which are designated as failures. (Note there is nothing pejorative in the use of the term
success or failure in this case.) Let x = the number of successes. Then the probability density
function of x is
Let Then it can be shown that and
Note that for very large N or very small n that these results approximate the binomial
distribution.
Example: Suppose that there are 25 persons in a class, 15 men and 10 women. Suppose that four
persons are chosen at random. Let x = the number of men among the four persons chosen. Find
P(x = 3) and find the mean and variance of x.
Solution.