Negative Hypergeometric Distribution

An experiment of sampling of objects without replacement till a fixed number of objects having a certain characteristic is obtained is known as negative hypergeometric experiment. Thus, a random variable X denoting the number of trials or drawings required to get a fixed number of objects with a specified characteristics is called negative hypergeometric random variable and its distribution is called negative hypergeometric distribution.

Defination

A random variable X is said to follow a negative hypergeometric ddistribution with three parameters P, Q, and r, if its probability mass function is given by

$$p(x; P, Q, r) \ = \ \frac{ \binom{x+r-1}{x} \ \binom{N-r-x}{Q-x}}{ \binom{N}{Q}} \ \ \ \ ; \ x \ = \ 0, \ 1, \ 2, ..., \ Q$$

where, N = P+Q and r∈ { 1, 2, 3, ..., P}.

Derivation of Negative Hypergeometric Distribution