Thread: X, Y iid, need moment generator function of sum and product

1. X, Y iid, need moment generator function of sum and product

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

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

2. I'll do the last one.

The MGF of 2 + 3X + 4Y, by definition is

$E(e^{(2 + 3X + 4Y)t})=e^{2t} E(e^{3Xt})E(e^{4Yt})$

by independence and algebra. Now combine the constants with the t's....

$=e^{2t} E(e^{X(3t)})E(e^{Y(4t)})$

$=e^{2t} M_X(3t)M_Y(4t)$

Finally plug in the 3t where you see a t into the MGF of X
and plug in the 4t where you see t in the MGF of Y.