# Joint probability of (X+Z) $\land$ (Y+Z) for arbitrary distribution of X, Y, Z.

I am trying to follow this answer and generalize it for the case of arbitrary probability distributions.

Say, we have three independent random variables $X$, $Y$, $Z$, with pdfs $f_X(x)$, $f_Y(y)$ and $f_Z(z)$. I'd like to find the joint pdf of $f_{X^{\prime}Y^{\prime}}(x^{\prime},y^{\prime})$ of $X^{\prime}=X+Z$ and $Y^{\prime}=Y+Z$. On the step, $$F_{X^{\prime}Y^{\prime}}=\mathbb{E}\left[F_X(x^{\prime}-Z)F_Y(y^{\prime}-Z)\right],$$ where $F$ denotes the cdf, I try to evaluate the expectation as: $$F_{X^{\prime}Y^{\prime}}=\int_{-\infty}^{\infty} F_X(x^{\prime}-z)F_Y(y^{\prime}-z)f_{Z}(z)dz,$$ which does not seem to work, since when I substitute any particular pdfs for $X$, $Y$ and $Z$, the resulting joint distribution just factorizes. What am I doing wrong?

• You can't find joint pdfs from marginal pdfs without making assumptions such as independence. – Dilip Sarwate Feb 7 '18 at 17:53
• I edited the question to take that into account, although I don't think it the reason of the original problem. – Ilya Feb 7 '18 at 18:23
• Can you give an example of the issue you are having? – owen88 Feb 7 '18 at 19:42
• I figured out the problem. When substituting the cdfs I was forgetting to set the x<0 and y<0 parts to 0. Once I took that into account, everything works. – Ilya Feb 8 '18 at 10:56