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

Question

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??

Answer 1

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: $$p_V(v) = \int p_{U,V}(u,v)\,\mathrm{d}u$$