Minimum of Independent Geometric Random Variables

Let $X_1, X_2, \ldots, X_n$ be independent, each $X_i \sim \text{Geometric}(p_i)$ with $P(X_i = k) = (1-p_i)^{k-1} p_i$ for $k = 1, 2, \ldots$ (the "number of trials until first success" convention).

Define $M = \min(X_1, \ldots, X_n)$.

(a) Show that $P(M > k) = \prod_{i=1}^{n} (1-p_i)^k$ for $k = 0, 1, 2, \ldots$

(b) Prove that $M \sim \text{Geometric}\!\left(1 - \prod_{i=1}^n (1-p_i)\right)$. State the PMF of $M$ explicitly.

(c) For the iid case $p_i = p$ for all $i$: express $E[M]$ and $\text{Var}(M)$ in terms of $n$ and $p$, and verify that $E[M] \to 1$ as $n \to \infty$.

(d) Application: Five independent traders each attempt to fill an order on any given day with probability $0.3$. What is the expected number of days until the first fill occurs? What is the probability that no fill occurs in the first 3 days?

(e) Show that $P(X_j = M \mid M = m)$ depends on $j$ (when $p_i$ are not all equal) and compute this probability for $n = 2$, $p_1 = 0.3$, $p_2 = 0.5$, $m = 2$.