Ideas of how to find the distribution of Gamma(N,$\lambda$), with N~Geom($\alpha$)

Question

I'm not exactly sure how to go about this, could I use the definition of the MGF or something along those lines?

Answer 1

Note that if $X\sim \text{Gamma}(N,\lambda)$ then $$ X\stackrel{d}{=}X_1+\dotsb+X_N $$ where $X_i\sim \text{Exp}(\lambda)$ are i.i.d exponential random variables with rate parameter equal to $\lambda$. It follows by the law of total expectation that $$ M_{X}(t)=Ee^{tX}=g_{N}(M_{X_1}(t)) $$ where $g_N$ is the probability generating function of $N$ and $M_{X_1}$ is the moment generating function of $X_1$.

You can do something similar using characteristic functions and then invert using the inversion formula.