Let $N$ be a non-negative integer-valued random variable with

$$P\{N=r\}=p_{r}, \quad r=0,1,2, \ldots$$

Define $\mathbb{E} N$, and show that

$$\mathbb{E} N=\sum_{n=1}^{\infty} P\{N \geqslant n\} .$$

Let $X_{1}, X_{2}, \ldots$ be a sequence of independent and identically distributed continuous random variables. Let the random variable $N$ mark the point at which the sequence stops decreasing: that is, $N \geqslant 2$ is such that

$$X_{1} \geqslant X_{2} \geqslant \ldots \geqslant X_{N-1}<X_{N},$$

where, if there is no such finite value of $N$, we set $N=\infty$. Compute $P\{N=r\}$, and show that $P\{N=\infty\}=0$. Determine $\mathbb{E} N$.