# How is the minimum of a random set of random variables distributed? [duplicate]

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?

• stats.stackexchange.com/questions/220/… Mar 22 '16 at 23:55
• 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. Mar 23 '16 at 0:22
• I'm not sure if this is really a duplicate of the linked thread, given the Poisson distribution of n. Mar 23 '16 at 0:26
• 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. Mar 23 '16 at 14:55
• 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: $$P(\min(X_1,\cdots,X_N)\leq x) = \sum_{n\geq 0} [1-(1-F(x))^n ] P(N=n)$$ Mar 23 '16 at 19:23