Specific formulae for some success run distributions

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Let N(k)n denote the number of success runs of length k (≥ 1) in n Bernoulli trials. A specific formula is derived for P(N(k)n = x) which is alternative to the one established by Philippou and Makri (1986) and Hirano (1986) and which is in a form suitable for the computation of asymptotic distributions (as in Godbole, 1990a, b); recall that N(k)n is said to have a binomial distribution of order k. In a similar fashion, different formulae are obtained for the geometric, negative binomial and Poisson distributions of order k (introduced by Philippou, Georghiou and Philippou, 1983. © 1990.

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Statistics and Probability Letters