Perhaps the most well known of the discrete distributions, the binomial distribution, we covered in Section 12.11. Equally well known, the Poisson distribution derives from the binomial distribution and was covered in Section 12.12. Here we describe the range of extensions to both.
The binomial distribution gives the probability of a number of successes; the negative binomial distribution gives the probability of a number of failures. Also known as the Pascal distribution, it is also a special case of the Pόlya distribution. It is a compound distribution, starting as the Poisson distribution. But, instead of the expected number of successes (λ) being a constant, it is assumed to follow the gamma distribution.
There are several ways of presenting the PMF. In the first, if p is the probability of success of a single trial, the PMF gives the probability of there being x failures before there are s successes. This means that there must first be (s − 1) successes and x failures, followed by a success. Therefore
The PMF can also be written in the form that gives the probability of there being x failures in n trials, where n will therefore be the sum of x and s.
In the batch blending example we require only one success. If ...