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JOINT DISTRIBUTIONS OF NUMBERS OF SUCCESS-RUNS AND

FAILURES UNTIL THE FIRST CONSECUTIVE *k* SUCCESSES

IN A BINARY SEQUENCE

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N. BALAKRISHNAN

*
Department of Mathematics and Statistics, McMaster University,*

Hamilton, Ontario, Canada L8S 4K1
(Received February 1, 1996; revised July 1, 1996)

**Abstract.**
Joint distributions of the
numbers of failures, successes and success-runs of
length less than *k* until the first consecutive *k*
successes in a binary sequence were derived recently
by Aki and Hirano (1995, *Ann.Inst.
Statist. Math.*, **47**, 225-235). In this
paper, we present an alternate derivation of these
results and also use this approach to establish some
additional results. Extensions of these results to
binary sequences of order *h* are also presented.

*Key words and phrases*:
Probability
generating function, waiting time, binary sequence of
order *k*, geometric distribution of order *k*.

**Source**
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