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
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
The Variance is

(6) 
Since is a Bernoulli variable,
so

(8) 
For , the Covariance is

(9) 
The probability that both and are successful for is
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,
Combining (11) with

(12) 
gives
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
so the final result is

(16) 
and, since

(17) 
and

(18) 
we have
The Skewness is
and the Kurtosis

(21) 
where
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. 532533, 1987.
Spiegel, M. R. Theory and Problems of Probability and Statistics.
New York: McGrawHill, pp. 113114, 1992.
© 19969 Eric W. Weisstein
19990525