Let X be a Poisson random variable with parameter 1. Show that P{X is even} = 3[1 + e^(-2)] by using the fact that the probability that the binomial random variable with parameters n and p is even is 1/2[1 + (q-p)^n], where q = 1-p, as stated in Theoretical Exercise 4.16, and the relationship between Poisson and binomial random variables.
(b) Verify the formula in part (a) directly by making use of the expansion of e^(-1) + e.