I'm trying to find the distribution and parameters in a Compound Poisson $S=\displaystyle\sum_{j=1}^{N}Y_{j},$ where $Y_{j}$ are exponential random variables independent and distributed identically with parameter $\alpha$ and $N$ is Poisson random variable with parameter $\lambda.$
I'm stuck because my attempts are based in computing the density probability function and I find a hard series: $$e^{-\lambda}e^{-\alpha x}\displaystyle\sum_{n=1}^{\infty}\frac{(\alpha\lambda)^{n}}{n!}\frac{x^{n-1}}{(n-1)!}.$$
The expression above was obteined using theorem of conditional probability in the random variable $N.$
If I use generating function I get $M_{S}(t)=e^{\lambda((\frac{\alpha}{\alpha-t})-1)},$ but I don't know the distribution of $S.$ This expression was got using conditional expectation again in random variable $N.$
Any kind of help is thanked in advanced.