An exam contains ten multiple choice probblems. Each problem only has one correct answer out of four choices. If a student randomly guess the answers for all the problems, what is the probability that he can pass the exam (at least guess six correctly)?

The number of people arriving at a library is a Poisson random variable. On average 10 customers arrive per hour. Let X be the number of people arriving from 9am to 10:30am. What is \(\mathbb{P} (10<X\leq 20)\)?

Let \(X\) be a discrete random variable. \(X=0,1,2,...\) Prove

\(\mathbb{E}X=\sum_{k=0}^{\infty}\mathbb(X>k)\)

Let \(X \sim Poisson(\lambda)\). Calculate the variance of \(X\)