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FORMULAE AND RECURSIONS FOR THE JOINT DISTRIBUTION

OF SUCCESS RUNS OF SEVERAL LENGTHS

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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 *p*_{j} of
success, 1 __<__ *j* __<__ *n*. Let *M*(*n*,*K*) and *N*(*n*,*K*) denote,
respectively, the *r*-dimensional random variables (*M*(*n*,k_{1}),
..., *M*(*n*,k_{r})) and (*N*(*n*,k_{1}), ..., *N*(*n*,k_{r})), where *K*=(k_{1},
k_{2}, ...,k_{r}) 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.

**Source**
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