Suppose that the moment generating function of a random variable $X$ is $M_X(t) = \mbox{exp}(2 e^t - 2)$ and that of a random variable $Y$ is $M_Y(t) = \left(\frac{7}{10}\, e^t + \frac{3}{10}\right)^{8}$. If $X$ and $Y$ are independent, find each of the following.

(a) $P\lbrace X + Y = 3 \rbrace =$

(b) $P\lbrace XY = 0 \rbrace =$

(c) $E[XY] =$

(d) $E[(X+Y)^2] =$