suppose $X_1,X_2,\ldots,X_n$ is random sample of $Exp(0,\sigma)$. if $\mathbb{S_n}=X_1+X_2+\ldots+X_n$ and $\mathbb{Z}=\max\{n:\mathbb{S_n}\leq s\}$ how can find distribution $\mathbb{Z}$?
finding distribution $\mathbb{Z}$ in problem
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probability
1 Answers
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Note that $(S_n)_{n\geqslant1}$ is the set of events of a Poisson process $(N_t)_{t\geqslant0}$ with intensity $\sigma$ and that $Z=N_s$ counts the number of events before time $s$. Hence, the distribution of $Z$ is Poisson with parameter $s\cdot\sigma$.