Let $\tau$ be a r.v. $\in(0,\infty)$ with PDF $f_\tau(\tau)=\lambda e^{-\lambda \tau}$. How do I find the PDF of $f_{\sum_{i=1}^NX_i}(\sum_{i=1}^NX_i)$ where $X=e^{-\tau}$?
I can easily find the PDF of $X$: $f_X(x)=f_\tau(-\log x)\frac{1}{x}\Theta(1-x)$ where $\Theta$ is the Heaviside function.
The characteristic function of the sum reads:
\begin{equation}
\phi_{\sum_{i=1}^NX_i}(s)=[E(e^{-sX})]^N=\left[\int dx\ e^{-sx} x^{\lambda-1}\Theta(1-x)\right]^N=[\lambda s^{-\lambda} (\Gamma[\lambda] - \Gamma[\lambda, s])]^N
\end{equation}
and now I should take the inverse Laplace transform in order to find the PDF of $\sum_{i=1}^NX_i$ but there is no way (I think) to find it.
Someone maybe know another way to find the PDF or maybe know how to perform the inverse Laplace transform?