Given $P_r\;{\raise.17ex\hbox{$\scriptstyle\sim$}}\;uniform(Pconst-a,Pconst+a)$
I want to modify $P_t$ in such a way, that it's range is symmetrically dependent on random variable $\xi \in [0,1]$ like this:
$P_r\;{\raise.17ex\hbox{$\scriptstyle\sim$}}\;uniform(\xi(Pconst-a),\xi(Pconst+a))$
PDF of $\xi$: $f_{\xi}(z) = \frac{2}{\pi \sqrt{1-z^2}},\; z \in [0,1]$
The problem is that I fail to express PDF of the modified $P_r$
I am trying to do it like this:
\begin{equation} f_{P_r | \xi}(z | y) = \begin{cases} \frac{1}{2a y}, \; \substack{ z \in \left[ y(P_{const}-a), y(P_{const}+a) \right] \\ y \in (0,1] }\\ 0, \; otherwise \end{cases} \label{eq:theProblem} \end{equation}
\begin{equation} f_{P_r}(z) = \begin{cases} \int_{0}^{1}{ f_{P_r | \xi}(z | y) f_{\xi}(y) } dy ,\; z \in \left[ P_{const}-a, P_{const}+a \right] \\ 0, \; otherwise \end{cases} \label{eq:res} \end{equation}
but I clearly need to use $f_{P_t, \xi}(z,y)$ in $f_{P_r | \xi}(z | y)$.
So the problem, I guess, boils down to me not knowing how to derive $f_{P_t, \xi}(z,y)$