On the number of trials needed to obtain consecutive successes
Introduction
We consider a well-known problem in applied probability in which independent Bernoulli trials, each having success probability , are performed until consecutive successes are achieved where . Let count the number of trials needed to obtain consecutive successes. Clearly, is a discrete random variable (rv) with probability mass function (pmf) on the support set . The distribution of has been studied previously, most notably by Shane (1973), who derived the probability generating function (pgf) of by developing a recursive formula for its pmf in terms of his Polynacci polynomials of order in . Other related papers followed, particularly those by Turner (1979), Philippou and Muwafi (1982), and Philippou et al. (1983). In the latter paper, the authors introduce a particular type of generalized geometric distribution to which the distribution of belongs (not surprisingly, given the fact that has a geometric distribution when ). Specifically, Philippou et al. (1983) show that where the above summation is over all non-negative integers such that Using the above pmf, Philippou et al. (1983) also derive the associated pgf as a means of obtaining (through differentiation) the following results for the mean and variance of : and For the sake of completeness, we remark that (2), (3) also agree with the results found independently by Woodside (1990).
In this paper, we present an alternative means of obtaining the pmf of , one which sheds a different light on the problem and ultimately gives rise to a simpler formula for which, unlike (1), does not involve the solutions of a diophantine equation. The approach we use is, in some sense, less specialized, and is based on a clever conditioning argument which Ross (2010, Example 3.15, p. 113) successfully employs to obtain . By conditioning on the rv (since one must first obtain consecutive successes before reaching ) and using the law of total expectation, Ross develops a recursive formula for which, when solved, agrees with (2) but does not involve any complicated sum formulas or the differentiation of a pgf. We adapt this argument to derive the pmf. An added advantage to our approach is that it leads to an equally elegant formula for the complementary cumulative distribution function (ccdf) of . Finally, we conclude our paper with a novel derivation revealing an interesting relationship between the factorial moments of and a sequence of polynomials with combinatorial significance, namely the exponential partial Bell polynomials.
Section snippets
Derivation of the pmf and ccdf of
We adopt the approach used by Ross (2010), as described above, but this time for the pmf. In particular, conditioning on the rv (for ), we first obtain since for . Now, for , we condition on the outcome of the trial. If this trial is a success, then with probability 1. However, if this trial is a failure, then the counter essentially resets itself following this failed trial.
Derivation of the factorial moments of
We now turn our attention to finding an expression for the th factorial moment of , given by Using (15), we have that Interchanging the order of summation in (20) leads to where . Applying Leibniz’s product rule for differentiation (and keeping in mind that the zeroth
Acknowledgments
The authors would like to thank the anonymous referee and Associate Editor for their helpful comments and useful suggestions. Steve Drekic would like to thank Zhuoqun Han for sparking initial interest into the investigation of this problem. Steve Drekic also acknowledges the financial support from the Natural Sciences and Engineering Research Council of Canada through its Discovery Grants program (RGPIN-2016-03685).
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