Reading the list of exercises about Poisson Process I came along to such task:
Let $(Sn; n \geq 1)$ be a partial sum of iid sequence with exponential distribution ($EX=\frac{1}{\lambda}$) and let $p_n$ be a density of variable $S_n$. How can we write down $p_{n+m}$ using $p_n$ and $p_m$. Discuss the importance of assumption that fucntion $x \mapsto p_n(x) $ is continous $(x \in R )$
I appreciate any hints, because I have no idea what should I do.
For any $n\in N$, $S_n \sim Erlang(n,\lambda)$
where $Erlang(k,\lambda)$ is the Erlang distribution, with pdf $ f(x;n,\lambda)= \frac{\lambda^nx^{n-1}e{-\lambda x}}{(n-1)!}$
The problem then just reduces to the property that if $X \sim Erlang(n,\lambda)$ and $Y \sim Erlang(m,\lambda)$ then, $X+Y \sim Erlang(n+m,\lambda)$. So all you need to do is to write the pdf of $S_n+S_m$ where $S_n$ and $S_m$ are independent(as they occur at different times), which is the convolution between them.