Find PDF of Z=X/(Y+c), c a constant and given independence of X and Y and the PDF of X and the PDF of Y

I want to find the PDF of $$Z=X/(Y+c)$$ where $$c$$ is a constant and $$X,Y$$ are two independent random variables. The PDFs of $$X$$ and $$Y$$ are supposed to be given. I would like to have a general form using the PDFs of $$X$$ and $$Y$$. $$X\geq 0$$ and $$Y\geq 0$$

I thought of using the CDF but I cannot write it in terms of the PDF of $$Y$$.

Any suggestions??

1 Answer

You should use a two variable transform and then marginalize out the second variable. A good resource on this kind of technique is the Casella and Berger book.

Use a two variable transform $$\varphi$$ as follows: $$\begin{bmatrix} U\\ V \end{bmatrix} = \varphi(X,Y) = \begin{bmatrix} X\\ X/(Y+c) \end{bmatrix}$$ Then $$\varphi$$ is invertible and its inverse is $$\begin{bmatrix} X\\ Y \end{bmatrix} = \varphi^{-1}(U,V) = \begin{bmatrix} \varphi^{-1}_1(U,V)\\ \varphi^{-1}_2(U,V) \end{bmatrix} = \begin{bmatrix} U\\ U/V - c \end{bmatrix}$$ The joint PDF of $$U,V$$ is $$p_{U,V}(u,v) = \lvert\det\mathrm{J}_{\varphi^{-1}}(u,v)\rvert p_{X,Y}(\varphi^{-1}_1(u,v), \varphi^{-1}_2(u,v))$$.

I'll let you compute the jacobian $$\mathrm{J}_{\varphi^{-1}}$$ and its determinant. Then you can obtain the marginal $$p_V$$ by marginalisation: $$p_V(v) = \int p_{U,V}(u,v)\,\mathrm{d}u$$