Let X1, …, Xm be independent nonnegative integer-valued random variables all having the same dis

Question

Ramesh Rajak · Accepted Answer

First, we note that $min(X_1, ldots, X_m) geq n$ if and only if all $X_i geq n$ for $i=1, ldots,

Let $X_{1}, \ldots, X_{m}$ be independent nonnegative integer-valued random variables all having the same distribution $\left\{p_{n}, n \geq 0\right\}$; and $r_{n}=$ $\sum_{k=n}^{\infty} p_{k} .$ Show that $$ E\left\{\min \left(X_{1}, \ldots, X_{m}\right)\right\}=\sum_{n=1}^{\infty} r_{n}^{m} $$