Statistics & Probability Letters 10 (1990) 119-124 North-Holland July 1990 zyxwvut SPECIFIC FORMULAE FOR SOME SUCCESS RUN DISTRIBUTIONS Anant P. GODBOLE Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA Received April 1989 Revised August 1989 Abstract: Let Nik) denote the number of success runs of length k ( > 1) in n Bernoulli trials. A specific formula is derived for P(N, ck) = 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,,(” 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. Keywords: Bernoulli trials, success runs of length k, discrete distributions of order k, occupancy models. 1. Introduction Let S,,, N/ k) and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA L, denote, respectively, the number of successes, the number of success runs of length k (a 1) and the length of the longest success run in n Bernoulli trials, each with success probability p, 0 <p < 1. Throughout, q will equal 1 -p. We shall, without exception, use the conventionally accepted method for counting runs (see Feller, 1968, p. 305, for details). The exact probability distribution of iVJk’ was derived by Philippou and Makri (1986) and also by Hirano (1986); they proved that (x=0, LL..,b/k]), 0) where the inner summation Cz is taken over all non-negative integers xi satisfying the condition xi + 2x, + * . . + kx, = n - i - kx. They call the probability distribution defined by (1) the binomial distribution of order k, noting that the k = 1 case corresponds to the ordinary binomial distribution. In a like fashion, Philippou and Makri (1986) showed that P( L, < k, S,, = r) =p’q”-’ C ‘$f,+x;;.‘.f;~-+:) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK (O<k<r<n)> O<iGk 2 where the inner summation C, is taken over all non-negative integers x, (1 <j < k + 1) satisfying xi +2x, + . . . +(k + 1)x,+, = n - i and x1 +x2 + . . . +xk+* = n - r, thereby obtaining a formula dif- ferent from the ones in Burr and Cane (1961) and Gibbons (1971), who had derived expressions for P( L, < k 1 S, = r) and P( L, = k 1 S,, = r) respectively. This work relates to Department of Navy Grant NOO014-89-J-1830 issued by the Office of Naval Research. The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein. 0167-7152/ 90/ $3.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland) 119