Hypergeometric Distribution

Similar to the Binomial Distribution but samples without replacement

probability of success is not constant from trial to trial
for the geometric distribution:
- finite population correction function

Probability Distribution Function

A random variable $X$ has a hypergeometric distribution $X \sim hypergeometric (n, N, M)$ if and only if its probability distribution is

P (X = x) = f (x; n, N, M) = \frac{(\begin{matrix} M \\ x \end{matrix}) (\begin{matrix} N - M \\ n - x \end{matrix})}{(\begin{matrix} N \\ n \end{matrix})}

for $x = 0, 1, 2, \dots, n,$ $x \leq n, x \leq M$ and $n - x \leq N - M$

$N =$ population size
$n =$ sample size
$M =$ size of one group in the population
$x =$ number of successes in one group among $n$ samples

\begin{aligned} E (X) & = \sum_{x = 0}^{n} x \frac{(\begin{array}{c} M \\ x \end{array}) (\begin{array}{c} N - M \\ n - x \end{array})}{(\begin{array}{c} N \\ n \end{array})} \\ = \frac{1}{(\binom{N}{n})} \sum_{x = 0}^{n} x \frac{M!}{x! (M - x)!} (\binom{N - M}{n - x}) \\ = \frac{1}{(\binom{N}{n})} \sum_{x = 1}^{n} \frac{x M (M - 1)!}{x (x - 1)! (M - 1 - (x - 1))!} (\binom{N - M}{n - x}) \\ = \frac{M}{(\binom{N}{n})} \sum_{x = 0}^{n - 1} \frac{(M - 1)!}{x! (M - 1 - x)!} (\binom{N - M}{n - (x + 1)}) \\ = \frac{M}{(\binom{N}{n})} \sum_{x = 0}^{n - 1} (\binom{M - 1}{x}) (\binom{N - M}{n - 1 - x}) \\ = \frac{M}{(\binom{N}{n})} (\binom{N - 1}{n - 1}) vandermondes \\ E (X) & = \frac{n M}{N} \end{aligned}

Counting Theorems#Vandermonde's (Zhu Shijie's) Identity

σ^{2} = \frac{n M (N - M) (N - n)}{N^{2} (N - 1)}

examples

taking 3 balls from a urn with 10 balls, we want only 2 red ones(4 red in total)

P (X = 2) = \frac{(\begin{matrix} 4 \\ 2 \end{matrix}) (\begin{matrix} 6 \\ 1 \end{matrix})}{(\begin{matrix} 10 \\ 3 \end{matrix})} = 0.3

with replacement (binomial)

P (X = 2) = (\begin{matrix} 3 \\ 2 \end{matrix}) 4^{2} {0.6}^{1} = 0.288