Can someone give me some direction on this? I tried something but I have no definitions or theorems to support whether what I did is correct. Thanks!

Let X~Normal($\displaystyle \mu$, $\displaystyle \sigma$), Y~Gamma($\displaystyle \alpha$, $\displaystyle \beta$), and X and Y are independent. The moment generating function of X and Y are

$\displaystyle m_{X}(t) = e^{\mu t+\sigma^2 t^2 /2}$ and $\displaystyle m_{Y}(t) = (1- \beta t)^\alpha$, respectively.

Find the moment generating function of X+Y.

Find the moment generating function of 5X.

Find the moment generating function of 2 + Y.

Find the moment generating function of 2 + 3X + 4Y.

Thanks!