# 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.

Conditional probability involving a geometric random variable
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 $M=max\left({X}_{1},...,{X}_{N}\right)$. Find $P\left(M\le x\right)$ by conditioning on N.
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Paige Paul
Step 1
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 $P\left(M\le x\right)$, however it seems to be easier to calculate $P\left(M\le x|N=k\right)$ and then calculate $P\left(M\le x\right)$.
Step 2
Note that: $P\left(M\le x\right)=\sum _{k=1}^{\mathrm{\infty }}P\left(M\le x,N=k\right)=\sum _{k=1}^{\mathrm{\infty }}P\left(N=k\right)P\left(M\le x|N=k\right)$