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Hypergeometric Distribution
المؤلف:
Beyer, W. H.
المصدر:
CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
...
18-4-2021
1680
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,
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(1) |
The probability of successful selections is then
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(2) |
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(3) |
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(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 | ![]() |
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The th selection has an equal likelihood of being in any trial, so the fraction of acceptable selections
is
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(5) |
i.e.,
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(6) |
The expectation value of is therefore simply
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(7) |
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(8) |
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(9) |
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(10) |
This can also be computed by direct summation as
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(11) |
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(12) |
The variance is
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(13) |
Since is a Bernoulli variable,
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
so
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(19) |
For , the covariance is
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(20) |
The probability that both and
are successful for
is
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(21) |
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(22) |
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(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,
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(24) |
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(25) |
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(26) |
Combining (26) with
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(27) |
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(28) |
gives
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(29) |
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(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
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(31) |
Combining equations (◇), (◇), (◇), and (◇) gives the variance
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(32) |
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(33) |
so the final result is
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(34) |
and, since
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(35) |
and
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(36) |
we have
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(37) |
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(38) |
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(39) |
This can also be computed directly from the sum
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(40) |
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(41) |
The skewness is
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(42) |
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(43) |
and the kurtosis excess is given by a complicated expression.
The generating function is
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(44) |
where is the hypergeometric function.
If the hypergeometric distribution is written
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(45) |
then
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(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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