Moment-generating function of $m$ independent variables

Question

Let $X_1,...,X_n$ be independent variables, each of them has a Discrete uniform distribution between $0$ and $m$, $m= \left( 2,3,4,,... \right)$.

Let $Y$ be a random variable which is defined by $Y = X_1 + X_2 +...+X_n$.

What is the Moment-generating function of $Y$?

I do know that since the $X_i$s are independent, the function is a sum of all the moment generating functions of the $X_i$s. I also know that each function is essentially the expected value of $e^{tx}$. However, here I'm lost.

Answer 1

The moment-generating function of a sum of independent random variables is the product of their moment generating functions.

To a large extent, that's the point of using the moment-generating function at all; the transform method turns the convolution that is the density/probability function of the sum into a pointwise product.

Answer 2

$Ee^{tY}=[\frac 1 {m+1} \frac {e^{t(m+1)}-1} {e^{t-1}}]^{n}$ for $t \neq 0$, $1$ for $t=1$. I have used the formula for a geometric sum as well as the fact that the MGF of a sum of independent random variables is the product of the individual MGF's.