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On the number of failures until a first success in \(n\) Bernoulli trials containing \(m=1,\dots,n\) successes. (English) Zbl 1038.90025
Summary: We describe a random variable \(D_{n,m}\), \(n\geq m\geq 1\), 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 \(D_{n,m}\). We define a random variable \(D_n\), \(n\geq 1\), to be the mean of \(D_{n,1}, \dots, D_{n,n}\). We show that \(E[D_n]\) 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.
90B25 Reliability, availability, maintenance, inspection in operations research