Consider independent discrete random variables $X_{1}, \ldots, X_{n}$ and assume $E\left[X_{i}\right]$ exists for all $i=1, \ldots, n$.

Show that

$$E\left[\prod_{i=1}^{n} X_{i}\right]=\prod_{i=1}^{n} E\left[X_{i}\right]$$

If the $X_{1}, \ldots, X_{n}$ are also positive, show that

$$\prod_{i=1}^{n} \sum_{m=0}^{\infty} P\left(X_{i}>m\right)=\sum_{m=0}^{\infty} P\left(\prod_{i=1}^{n} X_{i}>m\right)$$