Compound geometric distribution of order \(k\). (English) Zbl 1380.62064
Summary: The distribution of the number of trials until the first \(k\) consecutive successes in a sequence of Bernoulli trials with success probability \(p\) is known as geometric distribution of order \(k\). Let \(T_k\) be a random variable that follows a geometric distribution of order \(k\), and \(Y_{1},Y_{2},\dots \) a sequence of independent and identically distributed discrete random variables which are independent of \(T_k\). In the present article we develop some results on the distribution of the compound random variable \(S_{k} =\sum_{t=1}^{T_{k}}Y_{t}\).
MSC:
| 62E15 | Exact distribution theory in statistics |
| 60E05 | Probability distributions: general theory |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
compound distributions; geometric distribution of order \(k\); phase-type distribution; runsReferences:
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