Let $X_1,\ldots,X_n$ be identically and independently distributed as exponential with density $f(x)=e^{-x}$.
a. What is the moment generating function of $X_1$?
b. What is the moment generating function of $Y=\sum_{i=1}^n X_i$
c. What is the density function of $Y$?
For a. I get $M_{X_1}(t) = E(e^{X_1t})=\int_{-\infty}^\infty e^{X_1t}e^{-X_1}dX_1$, but this doesn't make sense since it would be infinite. Is there some way to tighten the bounds so that it becomes finite?
Then for b. if you plug it in the same way as $E(e^{Yt})$, you get a product of moment generating functions like the one above, but again if you integrate from negative infinity to infinity, it's not finite.
For c. I know you're somehow supposed to use the result from b. to derive the density function, but I am not exactly sure how.