negative binomial distribution

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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)/p2.

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 (PQ)n, where P=1/p and Q=(1− p)/p. Writing Y=Xn, 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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