The hypergeometric distribution arises when choosing "without replacement" from a finite population. Consider a population of size N which contains M elements having some property. We select a sample of size K from this population. The number in the sample that have this property is a hypergeometric random variable X.
The function ⌊x⌋ in the cdf denotes the "floor" or greatest integer function.
| Parameter | Range | Description |
|---|---|---|
| N | N = 0, 1, ... | Total size of population |
| M | M = 0, 1, ..., N | # elements with property |
| K | K = 0, 1, ..., N | # elements selected |
Probability Mass Function
Support
Mean
Variance
| Example | N | M | K |
|---|---|---|---|
| Eight cups of tea are made, four of which contain sugar. Four cups are chosen at random. Let X be the number chosen that contain sugar. | 8 | 4 | 4 |
| A standard deck of cards contains 52 cards total, of which 13 are hearts. Six cards are chosen at random. Let X be the number of hearts chosen. | 52 | 13 | 6 |
| A lottery selects 5 numbers out of 59 total. Let X be the number of matches between the numbers on a random ticket and the numbers drawn. | 59 | 5 | 5 |
X ∼ Hypergeometric(N, M, K)
E(X) = , Var(X) =
Note that:
- X ≤ M. Since only M elements have the given property, the maximum number of elements that can be chosen with the property is M.
- X ≥ K − (N − M). Since M elements have the given property, then N − M elements do not have the property. So, if K items are chosen, at least K − (N − M) items must have the property.
- The variance term (N − K)/(N − 1) is less than 1 for K > 1. Hence, the variance of a hypergeometric(N, M, K) distribution is less than that of the limiting binomial(K, M/N) distribution for K > 1.
Note that:
- X ≤ M. Since only M elements have the given property, the maximum number of elements that can be chosen with the property is M.
- X ≥ K − (N − M). Since M elements have the given property, then N − M elements do not have the property. So, if K items are chosen, at least K − (N − M) items must have the property.
- The variance term (N − K)/(N − 1) is less than 1 for K > 1. Hence, the variance of a hypergeometric(N, M, K) distribution is less than that of the limiting binomial(K, M/N) distribution for K > 1.
The illustration above shows a sample of K items chosen from a set of size N, of which M items have some property shown in green. The random variable X denotes the number of items chosen with this property, which has a hypergeometric(N, M, K) distribution.
The simulation above shows a sample of K items chosen from a set of size N, of which M items have some property. Items with the property are shown in green, and items without the property are shown in grey. The random variable X denotes the number of items chosen with this property, which has a hypergeometric(N, M, K) distribution. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =