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If $X_1,...,X_N$ are independent and identically distributed exponential random variables, what can be said about the distribution of $\text{min}(X_1,...,X_N)$ when $N$ is random and modelled as a Poisson random variable?

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    $\begingroup$ stats.stackexchange.com/questions/220/… $\endgroup$
    – Alex R.
    Mar 22 '16 at 23:55
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    $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$ Mar 23 '16 at 0:22
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    $\begingroup$ I'm not sure if this is really a duplicate of the linked thread, given the Poisson distribution of n. $\endgroup$ Mar 23 '16 at 0:26
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    $\begingroup$ No this question is not for course and it is not duplicated as In this case the $n$ is also random which is different from the other question that $n$ is fixed. $\endgroup$ Mar 23 '16 at 14:55
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    $\begingroup$ First you'll need to define the minimum when $N=0$ (or use a shifted poisson distribution starting at $N=1$). The random $N$ doesn't add much: Given $N=n$, you get $P(\min(X_1,\cdots,X_N)\leq x | N=n)=1-[1-F(x)]^n$. Notice that the minimum of exponential random variables is exponentially distributed with parameter $\lambda= \lambda_1+\cdots\lambda_n$. Now just sum over: $$P(\min(X_1,\cdots,X_N)\leq x) = \sum_{n\geq 0} [1-(1-F(x))^n ] P(N=n)$$ $\endgroup$
    – Alex R.
    Mar 23 '16 at 19:23