While modeling a problem I came upon the need to calculate
$$ (1)\ Pr(\sum_{i=1}^{n+1} x_i \leq S | \sum_{i=1}^{n} x_i \leq s)\\ x_i\sim Erlang(k,\lambda)\ i.i.d $$
My problem is that I have no idea how to approach calculating their joint distribution or conversely, how to go about proving their dependence or independence (talking about the two sums here, I know the distribution of each sum).
If we denote these sums $S_{n+1}$ and $S_{n}$ I'm aware that I can rewrite this probability as
$$ (2)\ \frac{Pr(-s\leq S_{n+1}-S_{n} \leq S\ \cap\ 0\leq S_{n+1} + S_{n} \leq\ S+s)}{Pr(S_{n} \leq\ s)} $$
Unfortunately re-framing the problem in this manner didn't help either, it led me to considering the following expresion
$$ (3)\ \frac{Pr(-s\leq x_{n+1} \leq S\ \cap\ 0\leq S_{n} \leq\ s)}{Pr(S_{n} \leq\ s)} $$
which is easily calculable, I'm not sure it follows from the previous expression, so if this is the correct direction I'd love to get an idea how to go from (2) to (3)
The characteristic function of of the Erlang Distribution is $$ CF = \left( {1 - {{it} \over \lambda }} \right)^{\, - k} $$
So the CF corresponding to the sum of $n$ i.i.d. erlang variables is $$ CF = \left( {1 - {{it} \over \lambda }} \right)^{\, - nk} $$ which tells that is corresponds to an Erlang variable with parameters $\lambda, \, nk$.
The PDF of the sum of $n$ variables is therefore $$ {{\gamma \left( {nk,\lambda x} \right)} \over {\left( {nk - 1} \right)!}} $$
Now $$ \eqalign{ & P\left( {\sum\limits_{k = 1}^{n + 1} {x_{\,k} } \le S\;\left| {\;\sum\limits_{k = 1}^n {x_{\,k} } \le s} \right.} \right) = \cr & = {{\int_{t = 0}^s {P\left( {x_{\,n} \le S - t \cap \left( {t \le \sum\limits_{k = 1}^n {x_{\,k} } \le t + dt} \right)} \right)} } \over {P\left( {\sum\limits_{k = 1}^n {x_{\,k} } \le s} \right)}} \cr} $$