hypergeometric distribution

Hypergeometric properties:

1. The experiment consists of drawing a random sample of size n without replacement and without regard to order from a collection of N objects.
2. Of the N objects, we want r , the other N-r do not have the trait.
3. The random variable $X$ denotes the number of objects obtained in n trials with the trait.

Definition ( Hypergeometric distribution):

$$f(x) = dfrac{left( begin{array} { c } { r } \ { x } end{array} right) left( begin{array} { c } { N - r } \ { n-x } end{array} right)}{left( begin{array} { c } { N } \ { n } end{array} right)} qquad qquad operatorname{max}[0, n- (N-r)] leq x leq operatorname{min}(n,r)$$

Expectation and Variance for the Hypergeometric Distribution:

$$E[X] = mu = np\ operatorname{Var} [X] = sigma^2 = npq dfrac{N-n}{N-1}$$

where $p = r/N$, $q = 1 - p$

Approximating the Hypergeometric Distribution:

If $n/N leq 0.05$, the hypergeometric distribution can be approximated by a binomial distribution with parameters $n$ and $p=r/N$.