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

单词

Negative Binomial Distribution

释义

Negative Binomial Distribution

Also known as the Pascal Distribution and Pólya Distribution. The probability of successes and failures in trials, and success on the th trial is

(1)

where

is a Binomial Coefficient. Let

			(2)
			(3)

The Characteristic Function is given by

(4)

and the Moment-Generating Function by

(5)

but, since

The Moments about zero

are therefore

(Beyer 1987, p. 487, apparently gives the Mean incorrectly.) The Moments about the mean are

The Mean, Variance, Skewness and Kurtosis are then

which can also be written

The first Cumulant is

(24)

and subsequent Cumulants are given by the recurrence relation

(25)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 533, 1987.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, p. 118, 1992.

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