# Hypergeometric Distribution: Definition, Properties and Application

### Hypergeometric Distribution

A discrete random variable X is said to have a  hypergeometric distribution if its probability density function is defined as,

$f\left&space;(&space;x;N,M,n&space;\right&space;)=\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}};&space;x=1,2,3,...,n$

where N is a positive integer , M is a non-negative integer that is at most N and n is the positive integer that at most M. If any distribution function is defined by the following probability function then the distribution is called hypergeometric distribution.

### Properties of Hypergeometric Distribution

• Hypergeometric distribution tends to binomial distribution if N➝∞ and K/N⟶p.
• Hypergeometric distribution is symmetric if p=1/2; positively skewed if p<1/2; negatively skewed if p>1/2.
• The mean of the hypergeometric distribution concides with the mean of the binomial distribution if M/N=p.

### Mean    Variance   CF    MGF        Application of Hypergeometric Distribution

• The hypergeometric distribution is commonly studied in most introductory probability courses. In introducing students to the hypergeometric distribution, drawing balls from an urn or selecting playing cards from a deck of cards are often discussed. This is a simple process which focus on sampling without replacement.
• We are also used hypergeometric distribution to estimate the number of fishes in a lake.