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. 532-533, 1987.
Spiegel, M. R. Theory and Problems of Probability and Statistics.
New York: McGraw-Hill, pp. 113-114, 1992.
© 1996-9 Eric W. Weisstein
1999-05-25