Find a expression using pgf $G_X$ for the probability that $X$ is divisible by 4.

Question

A random variable $X$ has probability generating function $G_X$.

(a) Find a simple expression using $G_X$ for the probability that $X$ is even. Hint: consider the value of $G_X(-1)$.

(b) Suggest a similar expression for the probability that $X$ is divisible by 4 -- be creative about what values of the generating function you might evaluate!

I had no problem with solving a simpler question in part (a), but am really struggling with part (b).

Answer 1

In part (a) you used two numbers such that both are square roots of 1. Can you extend this to something that involves 4? Hint: you need 4 numbers.

Find a, b, c, d, such that $$aG(i)+bG(-1)+cG(-i)+dG(1) = \sum_{4|n} p(n). $$