view the negative binomial as a distribution, where λ is itself a random variable, distributed as a gamma distribution with shape r and scale θ = or correspondingly rate β =. To display the intuition behind this statement, consider two independent Poisson processes, "Success" and "Failure", with intensities p and 1 − p. Together, the Success and Failure processes are equivalent to a single Poisson process of intensity 1, where an occurrence of the process is a success if a corresponding independent coin toss comes up heads with probability p; otherwise, it is a failure. If r is a counting number, the coin tosses show that the count of successes before the rth failure follows a negative binomial distribution with parameters r and p. The count is also, however, the count of the Success Poisson process at the random time T of the rth occurrence in the Failure Poisson process. The Success count follows a Poisson distribution with mean pT, where T is the waiting time for r occurrences in a Poisson process of intensity 1 − p, i.e., T is gamma-distributed with shape parameter r and intensity 1 − p. Thus, the negative binomial distribution is equivalent to a Poisson distribution with mean pT, where the random variate T is gamma-distributed with shape parameter r and intensity . The preceding paragraph follows, because λ = pT is gamma-distributed with shape parameter r and intensity . The following formal derivation (which does not depend on r being a counting number) confirms the intuition. Because of this, the negative binomial distribution is also known as the gamma–Poisson (mixture) distribution. The negative binomial distribution was originally derived as a limiting case of the gamma-Poisson distribution. Distribution of a sum of geometrically distributed random variables If Yr is a random variable following the negative binomial distribution with parameters r and p, and support {0, 1, 2, ...}, then Yr is a sum of r independent variables following the geometric distribution (on {0, 1, 2, ...}) with parameter p. As a result of the central limit theorem, Yr (properly scaled and shifted) is therefore approximately normal for sufficiently large r. Furthermore, if Bs+r is a random variable following the binomial distribution with parameters s + r and p, then In this sense, the negative binomial distribution is the "inverse" of the binomial distribution. The sum of independent negative-binomially distributed random variables r1 and r2