Let $X_1 , . . .$ be independent random variables with the common distribution function $F$, and suppose they are independent of $N$, a geometric random variable with parameter $p$. Let $M = max(X_1,…,X_N)$. Find $P(M \le x)$ by conditioning on $N$.
I'm pretty lost here. First of all, what does "the common distribution" refer to? Secondly, how do I "condition on $N$" ?
I know that $P(N=k)=p(1-p)^{k-1}$.
And I know that for two random variables, $A$ and $B$: $$P(A=a \mid B=b)=\frac{P(A=a \cap B=b)}{P(B=b)}$$.
I think I need to apply these formulas to the problem but am not sure how. Please let me know if you can help! Thanks!
Best Answer
The problem statement says that $X_1,X_2,...$ all have the same distribution.
Now if I understand the problem statement correctly, you should find $P(M\le x)$, however it seems to be easier to calculate $P(M \le x | N=k)$ and then calculate $P(M\le x)$. Note that:
$P(M\le x)=\sum\limits_{k=1}^{\infty}P(M \le x , N=k)=\sum\limits_{k=1}^{\infty}P(N=k)P(M \le x | N=k)$