I have two independent random variables X and Y such that the moment generating functions are respectively $G_X(t)= 1/3 + (2t)/3$ and $G_Y(t)= 1/4 + t/2 + t^2/4$.
If we consider $Z = X+Y$ . What would be the possible values of Z? and $P(Z=1)$?
I tried to find the moment generating function of Z with the product of $G_X(t)$ and $G_Y(t)$, but I didn't get the possible values nor P(Z=1).
Given that your functions are denoted by $G$ and they have no exponentials in them, you are not looking at moment generating functions but probability generating functions, and so $X$ and $Y$ can be read off from the coefficients: $$p_X(x)=\begin{cases}1/3&x=0\\2/3&x=1\end{cases}$$ $$p_Y(x)=\begin{cases}1/4&x=0\\1/2&x=1\\1/4&x=2\end{cases}$$ Thus the possible values of $Z$ are $\{0,1,2,3\}$ and $P(Z=1)=\frac13\frac12+\frac23\frac14=\frac13$.