Probability - Random Variables (Bernoulli trials,; Poisson and binomial random variables)

A First Course in Probability - Sheldon Ross (9th Edition)
Theoretical Exercises in Chapter 3: Conditional Probability and Independence and Chapter 4: Random Variables
E3.16: Independent trials that result in a success with probability p and a failure with probability 1-p are called Bernoulli trials. Let P_n denote the probability that n Bernoulli trials result in an even number of successes (0 being considered an even number). Show that
P_n = p(1 - P_{n-1}) + (1 - p) P_{n-1}, n \ge 1
and use this formula to prove (by induction) that
+_n = ( 1 + (1 - 2p)^n )/2.
E4.15: Suppose that n independent tosses of a coin having probability p of coming up heads are made. Show that the probability that an even number of heads results is (1 + (q-p)^n)/2, where q = 1-p. Do this by proving and then utilizing the identity
sum_{i=0}^[n/2] C(n, 2i) p^{2i} q^{n-2i} = ((p+q)^n + (q-p)^n)
where [n/2] is the largest integer less than or equal to n/2. Compare this exercise with E2.16.
E4.17: Let X be a Poisson random variable with parameter lambda.
(a) Show that
P(X is even) = (1 + e^{-2 lambda})/2
by using the result of E4.15 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^{- lambda} + e^{lambda}.