Let C be a non-negative integer-valued random variable with probability generating function E (s4) = 0(s) — e + e' e + e-1 (a) Calculate the pmf for C. That means calculate 11)(C = k) for every...

Let C be a non-negative integer-valued random variable with probability generating function
E (s4) = 0(s) — e + e' e + e-1
(a) Calculate the pmf for C. That means calculate 11)(C = k) for every integer k E N. Show all of your work and explain each step. (b) Calculate the expectation E(C). Show all of your work and explain each step. (c) Show that there exists a random variable X with generating func-tion
E (sx) = e(s)-1. Hint: Random sums of random variables.