Let $X$ be a random variable with generating function $G_X(s) = \frac{1}{2}(1+s)/(2-s)$. We toss a fair coin, then I want to find the generating function of a random variable $Z$ which is defined as $Z = X$ if the coin is head, and $Z = 2X$ if the coin is tail.
Could you help to see if my solution is correct? I do not think it is easy to simplify my final result, so I am not sure if it is correct or not.