Product of independent random variables following different distributions

Question

I need to find the CDF of the product of two independent random variables $Z=XY$. $X$ is defined in $\left ( -\infty,0 \right )$ and $\left ( 0, \infty \right)$. Y is defined in [$0,A_o^{2}$], whith $A_o$ a positive real number. The random variables $X$ and $Y$ follow different distributions. I try the standard CDF definition as shown in https://en.wikipedia.org/wiki/Product_distribution , but i seem to define wrong the integration limits. Given the definition range of the the random variables $X$ and $Y$ how should i define the CDF of $Z$?

Answer 1

Assuming that $X$ and $Y$ have densities $f_X$ and $f_Y$ we have $P(Z\leq t)=P(X\leq \frac t Y)=\int_0^{A_0^{2}} \int_{-\infty}^{t/y}f_X(u)f_Y(y)\, du\, dy$.