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Distributions related to \((k_{1},k_{2})\) events. (English) Zbl 1195.60024

Summary: Let \(Z_1,Z_2,\dots\) sequence of Bernoulli trials with success probability \(p=\text{Pr}\) \((Z_t=1)\) and failure probability \(q=\text{Pr}(Z_t=0)=1-p\), \(t\geq 1\). For positive integers \(k_1\) and \(k_2\) we consider the events \(E_1\): at least \(k_1\) consecutive 0’s are followed by at least \(k_2\) consecutive 1’s, \(E_2\): exactly \(k_1\) consecutive 0’s are followed by exactly \(k_2\) consecutive 1’s and \(E_3\): at most \(k_1\) consecutive 0’s are followed by at most \(k_2\) consecutive 1’s. Denote by \(X^{(i)}_n\) the number of occurrences of the event \(E_i\), \((i=1,2,3)\) in \(Z_1,Z_2,\dots,Z_n\) \((n\geq 1)\), and let \(T^{(i)}_r\) be the waiting time for the \(r\)-th occurrence of the event \(E_i\) \((i=1,2,3)\) in \(Z_1,Z_2,\dots\). In the present paper we employ the Markov chain embedding technique to derive exact formulas for the probability generating functions, the probability mass functions and the \(m\)-th moments \((m\geq 1)\) of \(X^{(i)}_n\) and \(T^{(i)}_r\) \((i=1,2,3)\). An application is also given.

MSC:

60E05 Probability distributions: general theory
60E10 Characteristic functions; other transforms
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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References:

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