Hypergeometric Distribution
Let there be
ways for a "good"
selection and
ways for a "bad"
selection 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)
| |||
![]() |
(3)
| ||
|
(4)
|
The hypergeometric distribution is implemented in the Wolfram Language as HypergeometricDistribution[N, n, m+n].
The problem of finding the probability of such a picking problem is sometimes called the "urn problem," since it asks for the probability that
out of
balls drawn are
"good" from an urn that contains
"good"
balls and
"bad" balls. It therefore
also describes the probability of obtaining exactly
correct balls in
a pick-
lottery from a reservoir of
balls (of which
are "good" and
are "bad").
For example, for
and
, the probabilities
of obtaining
correct balls are
given in the following table.
| number correct | probability | odds |
| 0 | 0.3048 | 2.280:1 |
| 1 | 0.4390 | 1.278:1 |
| 2 | 0.2110 | 3.738:1 |
| 3 | 0.04169 | 22.99:1 |
| 4 | 0.003350 | 297.5:1 |
| 5 | 10820:1 | |
| 6 |
The
th selection has an equal likelihood of
being in any trial, so the fraction of acceptable selections
is
|
(5)
|
i.e.,
|
(6)
|
The expectation value of
is therefore simply
|
(7)
| |||
|
(8)
| |||
|
(9)
| |||
|
(10)
|
This can also be computed by direct summation as
![]() |
(11)
| ||
|
(12)
|
The variance is
![]() |
(13)
|
Since
is a Bernoulli
variable,
|
(14)
| |||
|
(15)
| |||
|
(16)
| |||
|
(17)
| |||
|
(18)
|
so
|
(19)
|
For
, the covariance
is
|
(20)
|
The probability that both
and
are successful
for
is
|
(21)
| |||
|
(22)
| |||
|
(23)
|
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,
|
(24)
| |||
|
(25)
| |||
|
(26)
|
Combining (26) with
|
(27)
| |||
|
(28)
|
gives
|
(29)
| |||
|
(30)
|
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
![]() |
(31)
|
Combining equations (◇), (◇), (◇), and (◇) gives the variance
|
(32)
| |||
|
(33)
|
so the final result is
|
(34)
|
and, since
|
(35)
|
and
|
(36)
|
we have
|
(37)
| |||
|
(38)
| |||
|
(39)
|
This can also be computed directly from the sum
![]() |
(40)
| ||
|
(41)
|
The skewness is
|
(42)
| |||
|
(43)
|
and the kurtosis excess is given by a complicated expression.
The generating function is
![]() |
(44)
|
where
is the hypergeometric
function.
If the hypergeometric distribution is written
![]() |
(45)
|
then
|
(46)
|
where
is a constant.







hypergeometric distribution




