## Quick Reference

For trials classified as ‘success’ or ‘failure’, the distribution of *X*, the number of trials required in order to obtain *n* successes. The trials are presumed to be independent and it is assumed that each trial has the same probability of success, *p* (≠ 0 or 1). The probability function (*see diagram*) is The mean of this distribution is *n*/*p* and the variance is *n*(1−*p*)/*p*^{2}.

Special cases of the distribution were discussed by Pascal in 1679. The name ‘negative binomial’ arises because the probabilities are successive terms in the binomial expansion of (*P*−*Q*)^{−n}, where *P*=1/*p* and *Q*=(1− *p*)/*p*. Writing *Y*=*X*−*n*, an equivalent form for the distribution is The variable *X* may be regarded as the sum of *n* independent geometric variables, each with parameter *p*. The case *n*=1 therefore corresponds to the geometric distribution.

The arc-sinh transformation, suggested by Anscombe in 1948, results in a random variable *S* having an approximate standard normal distribution.

**Negative binomial distribution.** In each case *n*=5; the shape is dependent upon the value of *p*.

*Subjects:*
Probability and Statistics.

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