“Hypergeometric Distribution”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hypergeometric Distribution

释义

Hypergeometric Distribution

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 ofsuccessful selections,

(1)

The probability of

successful selections is then


			(2)

The

th selection has an equal likelihood of being in any trial, so the fraction of acceptable selections

(3)

(4)

The expectation value of


			(5)

The Variance is

(6)

Since

is a Bernoulli variable,



			(7)

(8)

For

, the Covariance is

(9)

The probability that both

and

are successful for



			(10)

But since

and

are random Bernoulli variables (each 0 or 1), their product isalso a Bernoulli variable. In order for

to be 1, both

and

must be1,



			(11)

Combining (11) with

(12)

gives



			(13)

There are a total of

terms in a double summation over

. However,

for

of these, so there are a totalof

terms in the Covariance summation

(14)

Combining equations (6), (8), (11), and (14) gives the Variance




			(15)

so the final result is

(16)

and, since

(17)

and

(18)

we have


			(19)

The Skewness is



			(20)

and the Kurtosis

(21)

where







			(22)

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.

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