Compound geometric distribution I'm given an exercise to find a distribution of a r.v. $S_N$ constructed as follows:
$$S_N = \sum_{k=1}^{N+1} X_k$$
where $N \sim Geom(p), X_k \sim Exp(\lambda)$ for $p \in(0,1), \lambda >0$.
I recall from Non-Life Insurance course the following property of Laplace transform/PGF:
$$L_{S_N}(t) = g_N(L_X(t))$$
Since $L_X(t) = \frac{\lambda}{\lambda+t}$ and $g_N(t) = \frac{p}{1-(1-p)t}$
, plugging in one expression into another I obtained:
$$L_{S_N}(t) = p+(1-p)\frac{p\lambda}{p\lambda+t}$$
So here is my actual question. Is it possible to somehow recover a cdf from the given Laplace transform of the corresponding distribution ? I found a script where it's done but simply as a "property" but I would like to know some more detail and the theory behind it.
Any hint would be highly appreciated :)
 A: Fixed $n$, the R.V. $S_{n+1}$ has Erlang density, i.e.,
\begin{equation}
\nonumber
f_{S_{n+1}} (t)
= \lambda^{n+1} \frac{t^n}{n!} e^{-\lambda t} 
\end{equation}
which implies
\begin{equation}
\nonumber
\Pr(S_{n+1}\le t)
= \int_0^t \lambda^{n+1} \frac{z^n}{n!} e^{-\lambda z}  \mbox{d}z.
\end{equation}
Therefore
$$
\Pr(S_N\le t)
= \Pr\left ( \bigcup_{n\ge 0} \left\{ S_N\le t \cap N=n\right\} \right)\\
= \sum_{n\ge 0}  \Pr\left ( S_{N}\le t \cap N=n \right) \\
= \sum_{n\ge 0}  \Pr\left ( S_{N}\le t| N=n\right) \Pr\left (N=n \right)\\
= \sum_{n\ge 0}  \Pr\left ( S_{n+1}\le t\right) \Pr\left (N=n \right)\\
= \sum_{n\ge 0} \int_0^t \lambda^{n+1} \frac{z^n}{n!} e^{-\lambda z} \mbox{d}z \,p^n (1-p)\\
= \lambda (1-p) \int_0^t \sum_{n\ge 0}  \frac{(p \lambda z)^n}{n!} e^{-\lambda z} \mbox{d}z  \\
= \lambda (1-p) \int_0^t  e^{p \lambda z}  e^{-\lambda z} \mbox{d}z \\
= \lambda (1-p) \int_0^t  e^{- \lambda(1-p) z} \mbox{d}z\\
= 1-  e^{- \lambda(1-p) t},
$$
for which we recognize that $S_n$ is a random variable with the exponential law of parameter $\lambda(1-p)$.
Note that I could invert the integral and summation because the involved terms are all positive.
The same conclusion can be obtained with a characteristic function argument.
