Let $X$ and $Y$ be two independent random variables where the moment generating function mgf are
$M_X(t) = e^{2(e^t -1)}$ and $M_Y(t) = \big( \frac{3e^t +1}{4}\big)^{10}$
what is the value of $P(X + Y = 2)$ and $E(XY)$ ?
For $E(XY) = E(X)~E(Y) $ since they are independent then $E(X) = M^{'}_X(0)$ so $$ E(XY) = E(X)~E(Y) = \big(2e^{0+e^0-2} \times \frac{15 e^0(3e^0+1)^9}{524288} \big) $$
For $P(X+Y = 2)$, I know that $P(X+Y=2) = P(X=0)P(Y=2)+P(X=1)P(Y=1)+P(X=2)P(Y=0)$ but I have a difficulty to continue! i.e to get the probabilities from the mgf, we need to use the inverse Laplace transform, but I have a difficulty to compute the integral $$ \frac{1}{2\pi i} \lim_{T \rightarrow \infty} \int_{\gamma - iT}^{\gamma + iT} e^{zx} E(e^{tZ})dz$$ any help plz