On the Number of Failures Until a First Success in n Bernoulli Trials Containing $m=1,\dots ,n$ Successes.

Abstract

We describe a random variable ${\text{D}}_{\text{{n,m}}}$, $\text{n}\ge \text{m}\ge 1$, as the number of failures until the first success in a sequence of n Bernoulli trials containing exactly m successes, for which all possible sequences containing m successes and n-m failures are equally likely. We give the probability density function, the expectation, and the variance of ${\text{D}}_{\text{{n,m}}}$. We define a random variable ${\text{D}}_{\text{n}}$, $\text{n}\ge 1$, to be the mean of ${\text{D}}_{\text{n,1}},\dots ,{\text{D}}_{\text{n,n}}$. We show that E$[{\text{D}}_{\text{n}}]$ is a monotonically increasing function of n and is bounded by $\ln$ n. We apply these results to a practical application involving a video-on-demand system with interleaved movie files and a delayed start protocol for keeping a balanced workload.