The negative binomial distribution, also known as the Pascal distribution or Pólya distribution, gives the probability of successes and failures in trials, and success on the th trial. The probability density function is therefore given by
(1)
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(2)
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(3)
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where is a binomial coefficient. The distribution function is then given by
(4)
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(5)
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(6)
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where is the gamma function, is a regularized hypergeometric function, and is a regularized beta function.
The negative binomial distribution is implemented in the Wolfram Language as NegativeBinomialDistribution[r, p].
Defining
(7)
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(8)
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the characteristic function is given by
(9)
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and the moment-generating function by
(10)
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Since ,
(11)
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(12)
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(13)
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(14)
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The raw moments are therefore
(15)
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(16)
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(17)
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(18)
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where
(19)
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and is the Pochhammer symbol. (Note that Beyer 1987, p. 487, apparently gives the mean incorrectly.)
This gives the central moments as
(20)
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(21)
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(22)
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The mean, variance, skewness and kurtosis excess are then
(23)
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(24)
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(25)
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(26)
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which can also be written
(27)
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(28)
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(29)
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(30)
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The first cumulant is
(31)
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and subsequent cumulants are given by the recurrence relation
(32)
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