## Hypergeometric Distribution

Let there be ways for a successful and ways for an unsuccessful trial out of a total of possibilities. Take samples and let equal 1 if selection is successful and 0 if it is not. Let be the total number of successful selections,

 (1)

The probability of successful selections is then

 (2)

The th selection has an equal likelihood of being in any trial, so the fraction of acceptable selections is
 (3)

 (4)

The expectation value of is
 (5)

The Variance is
 (6)

Since is a Bernoulli variable,
 (7)

so
 (8)

For , the Covariance is
 (9)

The probability that both and are successful for is
 (10)

But since and are random Bernoulli variables (each 0 or 1), their product is also a Bernoulli variable. In order for to be 1, both and must be 1,
 (11)

Combining (11) with
 (12)

gives
 (13)

There are a total of terms in a double summation over . However, for of these, so there are a total of terms in the Covariance summation
 (14)

Combining equations (6), (8), (11), and (14) gives the Variance
 (15)

so the final result is
 (16)

and, since
 (17)

and
 (18)

we have
 (19)

The Skewness is
 (20)

and the Kurtosis
 (21)

where
 (22)

The Generating Function is
 (23)

where is the Hypergeometric Function.

If the hypergeometric distribution is written

 (24)

then
 (25)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 532-533, 1987.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 113-114, 1992.