Hypergeometric Distribution
المؤلف:
Beyer, W. H.
المصدر:
CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
...
18-4-2021
1841
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
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
This can also be computed by direct summation as
The variance is
 |
(13)
|
Since 
is a Bernoulli variable,
so
 |
(19)
|
For 
, the covariance is
 |
(20)
|
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 (26) with
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
 |
(31)
|
Combining equations (◇), (◇), (◇), and (◇) gives the variance
so the final result is
 |
(34)
|
and, since
 |
(35)
|
and
 |
(36)
|
we have
This can also be computed directly from the sum
The skewness is
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.
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 532-533, 1987.
Feller, W. "The Hypergeometric Series." §2.6 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 41-45, 1968.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 113-114, 1992.
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