FORMULAE AND RECURSIONS FOR THE JOINT DISTRIBUTION
OF SUCCESS RUNS OF SEVERAL LENGTHS

ANANT P. GODBOLE 1, STAVROS G. PAPASTAVRIDIS 2 AND ROBERT S. WEISHAAR 3

1 Department of Mathematical Sciences, Michigan Technological University,
Houghton, MI 49931, U.S.A.
2 Department of Mathematics, University of Athens, Panepistemiopolis, 15710 Athens, Greece
3 Department of Mathematics, The Ohio State University, Columbus, OH 43210, U.S.A.

(Received November 15, 1993; revised March 18, 1996)

Abstract.    Consider a sequence of n independent Bernoulli trials with the j-th trial having probability pj of success, 1 < j < n. Let M(n,K) and N(n,K) denote, respectively, the r-dimensional random variables (M(n,k1), ..., M(n,kr)) and (N(n,k1), ..., N(n,kr)), where K=(k1, k2, ...,kr) and M(n,s) [N(n,s)] represents the number of overlapping [non-overlapping] success runs of length s. We obtain exact formulae and recursions for the probability distributions of M(n,K) and N(n,K). The techniques of proof employed include the inclusion-exclusion principle and generating function methodology. Our results have potential applications to statistical tests for randomness.

Key words and phrases:    Overlapping and non-overlapping success runs, distributions of order k, generating functions, tests for randomness, inclusion-exclusion.

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