I'm currently encountering a generelized $\chi^2$-distribution in my calculations.
I found the pdf of it online, but nothing given for the expectation value or anything else.
At the moment I'm trying to solve the following differential equation: $\dot{c}(t)=\int_0^t\mathrm{d}t_1Q(t-t_1)c(t_1)$
Here, $Q(t-t_1)=iY(t-t_1)+X(t-t_1)$
and $X(t-t_1)$ and $Y(t-t_1)$ have generalized $\chi^2$-distribution for each $t-t_1$.
How do i solve this, are there any approaches?
Thanks!
(The pdf I talked about is given by $f(x; k,\sigma_1^2,\ldots,\sigma_k^2) = \sum_{i=1}^k \frac{e^{-\frac x {\sigma_i^2}}}{\sigma_i^2 \prod_{j=1, j\neq i}^k \left(1- \frac{\sigma_j^2}{\sigma_i^2}\right)} \quad\text{for } x\geq0$)