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Negative Binomial Distribution

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The negative binomial distribution, also known as the Pascal distribution or Pólya distribution, gives the probability of r-1 successes and x failures in x+r-1 trials, and success on the (x+r)th trial. The probability density function is therefore given by

P_(r,p)(x)=p[(x+r-1; r-1)p^(r-1)(1-p)^([(x+r-1)-(r-1)])]
(1)
=[(x+r-1; r-1)p^(r-1)(1-p)^x]p
(2)
=(x+r-1; r-1)p^r(1-p)^x,
(3)

where (n; k) is a binomial coefficient. The distribution function is then given by

D(x)=sum_(n=0)^(x)(n+r-1; r-1)p^r(1-p)^n
(4)
=1-((1-p)^(x+1)p^rGamma(x+r+1)_2F^~_1(1,x+r+1;x+2;1-p))/(Gamma(r))
(5)
=I(p;r,x+1),
(6)

where Gamma(z) is the gamma function, _2F^~_1(a,b;c;z) is a regularized hypergeometric function, and I(z;a,b) is a regularized beta function.

The negative binomial distribution is implemented in the Wolfram Language as NegativeBinomialDistribution[r, p].

Defining

P=(1-p)/p
(7)
Q=1/p,
(8)

the characteristic function is given by

phi(t)=(Q-Pe^(it))^(-r),
(9)

and the moment-generating function by

M(t)=<e^(tx)>=sum_(x=0)^inftye^(tx)(x+r-1; r-1)p^r(1-p)^x.
(10)

Since (N; n)=(N; N-n),

M(t)=p^r[1-(1-p)e^t]^(-r)
(11)
M^'(t)=p^r(1-p)r[1-(1-p)e^t]^(-r-1)e^t
(12)
M^('')(t)=(1-p)rp^r(1-e^t+pe^t)^(-r-2)×(-1-e^tr+e^tpr)e^t
(13)
M^(''')(t)=(1-p)rp^r(1-e^t+e^tp)^(-r-3)×[1+e^t(1-p+3r-3pr)+r^2e^(2t)(1-p)^2]e^t.
(14)

The raw moments mu_n^'=M^((n))(0) are therefore

mu_1^'=(rq)/p
(15)
mu_2^'=(rq(1+rq))/(p^2)
(16)
mu_3^'=(q[rp^2+3pq(r)_1+q^2(r)_2])/(p^3)
(17)
mu_4^'=(q[rp^3+7p^2q(r)_1+6pq^2(r)_2+q^3(r)_3])/(p^4),
(18)

where

q=1-p
(19)

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

This gives the central moments as

mu_2=(r(1-p))/(p^2)
(20)
mu_3=(r(2-3p+p^2))/(p^3)=(r(p-1)(p-2))/(p^3)
(21)
mu_4=(r(1-p)(6-6p+p^2+3r-3pr))/(p^4).
(22)

The mean, variance, skewness and kurtosis excess are then

mu=(rq)/p
(23)
sigma^2=(rq)/(p^2)
(24)
gamma_1=(2-p)/(sqrt(rq))
(25)
gamma_2=(p^2-6p+6)/(rq),
(26)

which can also be written

mu=nP
(27)
sigma^2=nPQ
(28)
gamma_1=(Q+P)/(sqrt(rPQ))
(29)
gamma_2=(1+6PQ)/(rPQ)-3.
(30)

The first cumulant is

kappa_1=nP,
(31)

and subsequent cumulants are given by the recurrence relation

kappa_(r+1)=PQ(dkappa_r)/(dQ).
(32)

See also

Binomial Distribution

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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.

Referenced on Wolfram|Alpha

Negative Binomial Distribution

Cite this as:

Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NegativeBinomialDistribution.html

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