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JOINT DISTRIBUTIONS OF NUMBERS OF SUCCESS-RUNS AND

FAILURES UNTIL THE FIRST CONSECUTIVE *k* SUCCESSES

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SIGEO AKI^{1} AND KATUOMI HIRANO^{2}

^{1} *Department of Mathematical Science, Faculty of Engineering Science,*

Osaka University, Machikaneyama-cho, Toyonaka, Osaka 560, Japan

^{2} *The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan*
(Received May 20, 1994; revised October 12, 1994)

**Abstract.**
Joint distributions of the numbers of failures,
successes and success-runs of length less than *k* until the first
consecutive *k* successes are obtained for some random sequences such
as a sequence of independent and identically distributed integer
valued random variables, a {0,1}-valued Markov chain and a binary
sequence of order *k*. There are some ways of counting numbers of
runs with a specified length. This paper studies the joint
distributions based on three ways of counting numbers of runs, i.e.,
the number of overlapping runs with a specified length, the number of
non-overlapping runs with a specified length and the number of runs
with a specified length or more. Marginal distributions of them can
be derived immediately, and most of them are surprisingly simple.

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

**Source**
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