# Hypergeometric Distribution Formula

Hypergeometric distribution is a random variable of a hypergeometric probability distribution. Using the formula of you can find out almost all statistical measures such as mean, standard deviation, variance etc.

\[\large P(x|N,m,n)=\frac{\left(\binom{m}{x}\binom{N-m}{n-x}\right)}{\binom{N}{n}}\]

Where,

$N$: The number of items in the population.

$n$: The number of items in the sample.

$x$: The number of items in the sample that are classified as successes.

$P(x| N, n, k)$: hypergeometric probability – the probability that an n-trial hypergeometric experiment results in exactly $x$ successes, when the population consists of $N$ items, $k$ of which are classified as successes.

### Solved Examples

**Question 1: **Calculate the probability density function of the hypergeometric function if N, n and m are 50, 10 and 5 respectively ?

**Solution:
**Given parameters are,

N = 50

n = 10

m = 5

Formula for hypergeometric distribution is,

P(x|N,m,n) = $\frac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}}$

P(x|N,m,n) = $\frac{\binom{5}{x}\binom{50-5}{10-x}}{\binom{50}{10}}$

So, the probability distribution function is,

P(x|50, 5, 10) = {$\frac{\binom{5}{x}\binom{50-5}{10-x}}{\binom{50}{10}}$ $0\leq x\leq 10$ $}