Negative Binomial Distribution

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)
			(2)
			(3)

where  is a binomial coefficient. The distribution function is then given by

			(4)
			(5)
			(6)

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)
			(8)

the characteristic function is given by

(9)

and the moment-generating function by

(10)

Since ,

			(11)
			(12)
			(13)
			(14)

The raw moments  are therefore

			(15)
			(16)
			(17)
			(18)

where

(19)

and  is the Pochhammer symbol. (Note that Beyer 1987, p. 487, apparently gives the mean incorrectly.)

This gives the central moments as

			(20)
			(21)
			(22)

The mean, variance, skewness and kurtosis excess are then

			(23)
			(24)
			(25)
			(26)

which can also be written

			(27)
			(28)
			(29)
			(30)

The first cumulant is

(31)

and subsequent cumulants are given by the recurrence relation

(32)

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.