Let $ R$ , $ C$ , and $ X$ be independent random variables defined on $ (0,\infty)$ and $ $ Y=\underbrace{R\, X}_{Z}+C.$ $ We are given the joint probability distribution of $ X$ and $ Y$ , $ P_{XY}(x,y)$ and are asked to calculate the probability distributions of $ R$ and $ C$ .

This is kind of like a regression problem, except I want the full probability distributions for the slope and intercept, not just their mean.

Here is what I have so far $ $ \begin{align} P_{XY}(x,y) &= P_X(x)P_Y(y|x)\ &= P_X(x)\int_0^\infty P_C(c)P_Z(y-c|x)dc\ &= P_X(x)\int_0^\infty P_C(c)\frac1xP_R\left(\frac{y-c}{x}\right)dc\ &= \frac{P_X(x)}{x}\int_0^\infty P_C(c)P_R\left(\frac{y-c}{x}\right)dc \end{align}$ $ Therefore, $ $ \frac{x\, P_{XY}(x,y)}{P_X(x)} = \int_0^\infty P_C(c)\,P_R\left(\frac{y-c}{x}\right)dc.$ $ The right hand side is something like a convolution (not quite), and its value is known for every pair of x and y. How do I find $ P_C$ and $ P_R$ ? Any hints for analytical or numerical solution will be appreciated.

I am reposting this from stackexchange: https://math.stackexchange.com/q/2541446/491395

Edit: As Bjørn’s answer below shows, we need more assumptions for this to work. Here is what I’m trying to do: I have measured the joint probability distribution of $ X$ and $ Y$ and it looks like this

Assuming a linear model with random slope and intercept works on this data, I want to find the distribution of these slopes and intercepts. Not sure, exactly what the necessary and sufficient conditions are for this to be possible.