Question regarding expected value of "composition"

Question

The question is to calculate $Emin[X_1,...X_{N+1}]$, where $N,X_1,...$ are independent, and $N$ has poisson distribution with parameter 1, and $X_i$ has expotential distribution with parameter 2. My question is, do I first calculate the expectation of $N$, then put it to my expected value and calculate it? I know how to do it, but it is the first time I have meet with such "composition" and I am not sure on how to approach it.

Answer 1

You cannot just replace $N$ with its expected value inside the expectation. However, you can condition on $N$ and sum over all its values, that is, you can say that $$\mathbb{E}[\min\{X_1, X_2,\ldots X_{N+1}\}] = \sum\limits_{n=0}^{\infty}\mathbb{E}[\min\{X_1, X_2,\ldots X_{N+1}\}\mid N = n] \mathbb{P}(N=n).$$

See the law of total expectation.