題組內容

3. (5% each, totally 15%) Let X and Y are two independent random variables, with the PMFs as follow. $$P_X(x) = \begin{cases} 1/2 & \text{if } x=1,2; \\ 0 & \text{else}. \end{cases}$$ $$P_Y(y) = \begin{cases} 1/2 & \text{if } x=3,4; \\ 0 & \text{else}. \end{cases}$$

(b) Let \( Z = X + Y \). What is the moment generating function of \( Z \)?