I had some trouble working on some problems and I would appreciate some help.
1), Let $\sum_{i=1}^{N}X_i=S$ where each X are iid and N is a random variable independent from all Xs. My goal is to figure out what $E[\bar X]$ is using the identity in the title.
I understand that $$E[\bar X]=E\left[\frac{S}{N}\right]$$ but I am not sure how to proceed from here.
I don't want to think that $1\over N$ comes out...
2), Finding the pdf for the minimum of the order statistics. Intuitively, $$\frac{d}{dx}[1-F(x)]^n$$ is what we use, but if we take the derivative, due to chain rule $-f(x)$ comes out, so the formula becomes
$$f_{X_{(1)}}=n[1-F(x)]^{n-1}(-f(x))$$
I do not know how the minus sign supposed to cancel...
I appreciate your help.