Exponential Expectation (a) Let X ∼ Exp(λ). Use induction to show that E[X k ] = k!/λ k for every k ∈ N. (b) For any |t| < λ, compute E[e tX] directly from the definition of expectation. (c) Using part (a), compute ∑ [infinity] k=0 E[X k ] k! t k . (d) Let M(t) = E[e tX] be a function defined for all t such that |t| < λ. What is dM(t) dt t=0 ? What is d 2M(t) dt2 t=0 ? How does each of these relate to the mean and variance of an Exp(λ) distribution?