Identities and Inequalities with Expectation For this exercise, the following identity might be useful: for a probability event A, P(A) = E[1{A}], where 1{·} is the indicator function. 1. Let X be a random variable with density f(x) = λe −λx 1{x > 0}. Show that E[X k ] = k!/λk for integer k ≥ 0. Hint: One way to attempt this problem is by using induction on k.