with the same value for parameter p is negative-binomially distributed with the same p but with r-value r1 + r2. The negative binomial distribution is infinitely divisible, i.e., if Y has a negative binomial distribution, then for any positive integer n, there exist independent identically distributed random variables Y1, ..., Yn whose sum has the same distribution that Y has. Representation as compound Poisson distribution The negative binomial distribution NB(r,p) can be represented as a compound Poisson distribution: Let denote a sequence of independent and identically distributed random variables, each one having the logarithmic distribution Log(p), with probability mass function Let N be a random variable, independent of the sequence, and suppose that N has a Poisson distribution with mean . Then the random sum is NB(r,p)-distributed. To prove this, we calculate the probability generating function GX of X, which is the composition of the probability generating functions GN and GY1. Using and we obtain which is the probability generating function of the NB(r,p) distribution. The following table describes four distributions related to the number of successes in a sequence of draws: class of distributions The negative binomial, along with the Poisson and binomial distributions, is a member of the (a,b,0) class of distributions. All three of these distributions are special cases of the Panjer distribution. They are also members of a natural exponential family. Statistical inference Parameter estimation MVUE for p Suppose p is unknown and an experiment is conducted where it is decided ahead of time that sampling will continue until r successes are found. A sufficient statistic for the experiment is k, the number of failures. In estimating p, the minimum variance unbiased estimator is Maximum likelihood estimation When r is known, the maximum likelihood estimate of p is but this is a biased estimate. Its inverse (r + k)/r, is an unbiased estimate of 1/p, however. When r is unknown, the maximum likelihood estimator for p and r together only exists for samples for which the sample variance is larger than the sample mean. The likelihood function for N iid observations (k1, ..., kN) is from which we calculate the log-likelihood function To find the maximum we take the partial derivatives with respect to r and p and set them equal to zero: and where is the digamma function. Solving the first equation for p gives: Substituting this in the second equation gives: This equation cannot