Let X be a Poisson random variable with parameter . (a) Show that P{X is even} = 1/2[1 + e2] by using the result of

Let X be a Poisson random variable with parameter λ.
(a) Show that
P{X is even} = 1/2[1 + e−2λ]
by using the result of Theoretical Exercise 15 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−λ + eλ.