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ON A WAITING TIME DISTRIBUTION IN A SEQUENCE

OF BERNOULLI TRIALS

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M. V. KOUTRAS

*Department of Mathematics, University of Athens,
Panepistemiopolis, 15784, Greece*
(Received June 27, 1995; revised November 6, 1995)

**Abstract.**
In the present article we investigate the exact
distribution of the waiting time for the *r*-th non-overlapping
appearance of a pair of successes separated by at most *k* - 2 failures
(*k* __>__ 2) in a sequence of independent and identically distributed
(iid) Bernoulli trials. Formulae are provided for the probability
distribution function, probability generating function and moments
and some asymptotic results are discussed. Expressions in terms of
certain generalised Fibonacci numbers and polynomials are also
included.

*Key words and phrases*:
Waiting times, success runs,
scans, distributions of order *k*, Fibonacci numbers, unimodality,
Markov dependence.

**Source**
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