Quote:

Find $\displaystyle P(M\leq x)$ by conditioning on N

At the first glance I didn't quite understand the information given in this problem. But after a while I wrote down what I believe is asked to solve in the problem

$\displaystyle P(M\leq x)=\sum_{k=0}^{\infty} P(M\leq x|N=k)P(k) $

But this is as far as I can go, I know P(k) = p(1-p)^(k-1), and I'm clueless on what's $\displaystyle P(M\leq x|N=k) $

I want to say that $\displaystyle M\leq x $ is independ of $\displaystyle N=k $, but I'm not sure whether it's true.

Since $\displaystyle X_1,X_2,\ldots$ are independent of $\displaystyle N$, conditioning by $\displaystyle N=k$ just gives $\displaystyle P(\max(X_1,\ldots,X_N)\leq x|N=k)=P(\max(X_1,\ldots,X_k)\leq x)$ (the conditioning doesn't affect the $\displaystyle X_i$'s by independence; it just fixes the value of $\displaystyle N$).