Hypergeometric distribution

The hypergeometric distribution describes the probability of a sample of size containing successes, drawn from a pool consisting of successes and failures, without replacement. #m/def/prob It has the probability mass function

Proof

We draw times from a pool of size , so the total number of outcomes is . Using the naïve definition of probability, the number of outcomes with will be equal to the number of ways of choosing of successes and of failures, giving .

Properties

Let . Let , , and .

Expectation:
Variance:

Proof of 1–2

Let be the indicator random variable for the th draw being a success, so that . It follows and hence , proving ^P1. We also have . Notice that by symmetry, for . Now

where

proving ^P2.

Furthermore