Distribution of the Maximum of Independent Geometric Random Variables

Let $X_1, X_2, \ldots, X_n$ be independent $\text{Geometric}(p)$ random variables with $P(X_i = k) = (1-p)^{k-1} p$ for $k = 1, 2, \ldots$ Define $M = \max(X_1, \ldots, X_n)$.

(a) Show that $P(M \le m) = [1 - (1-p)^m]^n$ for $m = 1, 2, \ldots$

(b) Derive $P(M = m)$ from the CDF.

(c) Express $E[M]$ as an infinite series using the tail-sum formula $E[M] = \sum_{m=0}^{\infty} P(M > m)$. Simplify to: $$E[M] = \sum_{m=0}^{\infty} \left[1 - (1 - (1-p)^m)^n\right].$$

(d) For the special case $n = 2$, $p = 1/2$, compute $P(M = 1)$, $P(M = 2)$, $P(M = 3)$ and verify they sum to nearly 1. Compute $E[M]$ exactly by evaluating the series.

(e) For general $n$ and small $p$, argue heuristically that $E[M] \approx \frac{\ln n}{p}$ by comparing to the continuous exponential analogue.