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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.

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Your sollution is overly complicated. The easy path to finding the probability generating function of the product of independent random variables is to use the definition.

$$\mathsf G_X(t)=\mathsf E(t^X)$$

Let $Y$ be the random variable resulting in $1$ if the coin toss is head and $2$ if it is tails. So $Z=YX$ and $Y\sim\mathcal U\{1,2\}$.

$$\begin{align}\mathsf G_Z(s)&=\mathsf E(s^Z)\\&=\mathsf E(s^{YX})\\&=\mathsf E(\mathsf E(s^{YX}\mid Y))&&\text{Law of Total Expectation}\\&=\mathsf E(\mathsf E(s^{YX}))&&\text{Independence}\\&=\mathsf E(\mathsf E({(s^Y)}^X))\\&~~\vdots\end{align}$$

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