k, provided r is an integer. (If r is a negative non-integer, so that the exponent is a positive non-integer, then some of the terms in the sum above are negative, so we do not have a probability distribution on the set of all nonnegative integers.) Now we also allow non-integer values of r. Then we have a proper negative binomial distribution, which is a generalization of the Pascal distribution, which coincides with the Pascal distribution when r happens to be a positive integer. Recall from above that The sum of independent negative-binomially distributed random variables r1 and r2 with the same value for parameter p is negative-binomially distributed with the same p but with r-value r1 + r2. This property persists when the definition is thus generalized, and affords a quick way to see that the negative binomial distribution is infinitely divisible. Recurrence relation The following recurrence relation holds: Related distributions The geometric distribution (on { 0, 1, 2, 3, ... }) is a special case of the negative binomial distribution, with The negative binomial distribution is a special case of the discrete phase-type distribution. The negative binomial distribution is a special case of discrete compound Poisson distribution. Poisson distribution Consider a sequence of negative binomial random variables where the stopping parameter r goes to infinity, while the probability p of success in each trial goes to one, in such a way as to keep the mean of the distribution (i.e. the expected number of failures) constant. Denoting this mean as λ, the parameter p will be p = r/(r + λ) Under this parametrization the probability mass function will be Now if we consider the limit as r → ∞, the second factor will converge to one, and the third to the exponent function: which is the mass function of a Poisson-distributed random variable with expected value λ. In other words, the alternatively parameterized negative binomial distribution converges to the Poisson distribution and r controls the deviation from the Poisson. This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Poisson for large r, but which has larger variance than the Poisson for small r. Gamma–Poisson mixture The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can