Negative Binomial Distribution

Also known as the Pascal Distribution and Pólya Distribution. The probability of $r-1$ successes and $x$ failures in $x+r-1$ trials, and success on the $(x+r)$th trial is

$$p\left[{{x+r-1\choose r-1} p^{r-1}(1-p)^{[(x+r-1)-(r-1)]}}\r... ...se r-1}p^{r-1}(1-p)^x}\right]p = {x+r-1\choose r-1}p^r(1-p)^x,$$ (1)

where ${n\choose k}$ is a Binomial Coefficient. Let

$$P = {1-p\over p}$$ (2)

$$Q = {1\over p}.$$ (3)

The Characteristic Function is given by

$$\phi(t)=(Q-Pe^{it})^{-r},$$ (4)

and the Moment-Generating Function by

$$M(t) = \langle e^{tx}\rangle = \sum_{x=0}^\infty e^{tx}{x+r-1\choose r-1}p^r(1-p)^x,$$ (5)

but, since ${N\choose n}={N\choose N-m}$,

$$M(t) = p^r \sum_{x=0}^\infty{x+r-1\choose x}[(1-p)e^t]^x$$

$$= p^r[1-(1-p)e^t]^{-r}$$ (6)

$$M'(t) = p^{r}(-r)[1-(1-p)e^t]^{-r-1}(p-1)e^t$$

$$= p^{r}(1-p)r[1-(1-p)e^t]^{-r-1}e^t$$ (7)

$$M''(t) = (1-p)r p^r(1-e^t+p e^t)^{-r-2}(-1-e^t r+e^t pr)e^t$$ (8)

$$M'''(t) = (1-p)r p^r(1-e^t+e^t p)^{-r-3}[1+e^t(1-p+3r-3pr)+r^2 e^{2t}(1-p)^2]e^t.$$ (9)

The Moments about zero $\mu_n'=M^{n}(0)$ are therefore

$$\mu_1' = \mu={r(1-p)\over p}={rq\over p}$$ (10)

$$\mu_2' = {r(1-p)[1-r(p-1)]\over p^2}={rq(1-rq)\over p^2}$$ (11)

$$\mu_3' = {(1-p)r(2-p+3r-3pr+r^2-2pr^2+p^2r^2\over p^3}$$ (12)

$$\mu_4' = {(-1 + p) r (-6 + 6 p - p^2 - 11 r + 15 p r - 4 p^2r - 6 r^2\over p^4}$$

$$\phantom{=} + {12 p r^2 - 6 p^2 r^2 - r^3 + 3 p r^3 - 3 p^2 r^3 + p^3 r^3)\over p^4}.$$ (13)

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

$$\mu_2 = \sigma^2 = {r(1-p)\over p^2}$$ (14)

$$\mu_3 = {r(2-3p+p^2)\over p^3}={r(p-1)(p-2)\over p^3}$$ (15)

$$\mu_4 = {r(1-p)(6-6p+p^2+3r-3pr)\over p^4}.$$ (16)

The Mean, Variance, Skewness and Kurtosis are then

$$\mu = {r(1-p)\over p}$$ (17)

$$\gamma_1 = {\mu_3\over\sigma^3}={r(p-1)(p-2)\over p^3} \left[{p^2\over r(1-p)}\right]^{3/2}$$

$$= {r(2-p)(1-p)\over p^3} {p^3\over r(1-p)\sqrt{1-p}}$$

$$= {2-p\over\sqrt{r(1-p)}}$$ (18)

$$\gamma_2 = {\mu_4\over\sigma^4}-3$$

$$= {-6 + 6p - p^2 - 3 r + 3p r\over (p-1) r},$$ (19)

which can also be written

$$\mu = nP$$ (20)

$$\mu_2 = nPQ$$ (21)

$$\gamma_1 = {Q+P\over\sqrt{rPQ}}$$ (22)

$$\gamma_2 = {1+6PQ\over rPQ}-3.$$ (23)

The first Cumulant is

$$\kappa_1=nP,$$ (24)

and subsequent Cumulants are given by the recurrence relation

$$\kappa_{r+1}=PQ{d\kappa_r\over dQ}.$$ (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.