Hypergeometric random variables

statistics

Hypergeometric random variables have these characteristics:

1. The experiment consists of drawing $n$ elements without replacement from a set of $N$ elements, $r$ of which are S (success) and $(N-r)$ of which are F (failure).
2. The hypergeometric random variable x is the number of S’s draw of n elements.

This is unlike bernouli distribution, because these outcomes are dependent upon one another. Assuming the first draw is odds of $r/N$, then the next draw will either be $\frac{r-1}{N-1}$ or $\frac{r}{N-1}$.

Probability distribution, mean, variance

$$\begin{array}{r}p(x)=\frac{\left(\genfrac{}{}{0ex}{}{r}{x}\right)\left(\genfrac{}{}{0ex}{}{N-r}{n-x}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}\end{array}$$

x = maximum [0, n-(N-r)],… minimum (r, n)

$$\begin{array}{r}\mu =\frac{nr}{N}\\ \\ {\sigma }^2=\frac{r(N-r)n(N-n)}{N^2(N-1)}\end{array}$$

N : number of elements

r : number of successes

n : number drawn

x : number of successes drawn in the n elements.