Probability problem with Setting up Fz(z) with the jointly continuous density

Question

Suppose that X and Y are are jointly continuous and have density f(x, y) . Let Z = Y / X ; we want the density of Z. a) Write down the expression for F_{Z}(z) , realizing that you have to be careful about y / x <= z turning into something more convenient because may be positive or negative, making it necessary to split the double integrals.

Answer 1

By definition, $F_Z(z) = Pr(Z \leq z) = Pr(Y/X \leq z)$. There are multiple cases. If $X<0$, we want $Pr(Y \geq Xz)$, while if $X > 0$ we want $Pr(Y \leq Xz)$. Combining these two pieces we get $$\int_{-\infty}^0\int_{xz}^\infty f(x, y) dydx + \int_{0}^\infty\int_{-\infty}^{xz} f(x, y) dydx.$$

Note that the first integral integrates over $x < 0$, and the second integrates over $x > 0$. It is not clear to me what you mean by "you have to be careful about y / x <= z turning into something more convenient", but my best guess is that you are referring to this issue.