Paper 2, Section I, F

Probability | Part IA, 2014

Consider independent discrete random variables X1,,XnX_{1}, \ldots, X_{n} and assume E[Xi]E\left[X_{i}\right] exists for all i=1,,ni=1, \ldots, n.

Show that

E[i=1nXi]=i=1nE[Xi]E\left[\prod_{i=1}^{n} X_{i}\right]=\prod_{i=1}^{n} E\left[X_{i}\right]

If the X1,,XnX_{1}, \ldots, X_{n} are also positive, show that

i=1nm=0P(Xi>m)=m=0P(i=1nXi>m)\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)

Typos? Please submit corrections to this page on GitHub.