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DISTRIBUTIONS OF NUMBERS OF FAILURES AND SUCCESSES

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 March 15, 1993; revised June 21, 1993)

**Abstract.**
Exact distributions of the numbers of
failures, successes and successes with indices no less than *l*
(1 __<__ *l* __<__ *k*-1) until the first consecutive *k* successes
are obtained for some {0, 1}-valued random sequences such as
a sequence of independent and identically distributed (iid)
trials, a homogeneous Markov chain and a binary sequence of order
*k*. The number of failures until the first consecutive *k*
successes follows the geometric distribution with an appropriate
parameter for each of the above three cases. When the
{0, 1}-sequence is an iid sequence or a Markov chain, the
distribution of the number of successes with indices no less
than *l* is shown to be a shifted geometric distribution of
order *k-l*. When the {0, 1}-sequence is a binary sequence of
order *k*, the corresponding number follows a shifted version of
an extended geometric distribution of order *k-l*.

*Key words and phrases*:
Geometric distribution,
discrete distributions, Markov chain, waiting time, geometric
distribution of order *k*, iid sequence, binary sequence of
order *k*, inverse sampling.

**Source**
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