# Product of independent random variables following different distributions

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 $$Z$$?

Assuming that $$X$$ and $$Y$$ have densities $$f_X$$ and $$f_Y$$ we have $$P(Z\leq t)=P(X\leq \frac t Y)=\int_0^{A_0^{2}} \int_{-\infty}^{t/y}f_X(u)f_Y(y)\, du\, dy$$.
• Thank you for your answer Mr. Murthy. Maybe i did not make clear that that $X$ is defined as a piecewise function in the intervals $\left ( -\infty,0 \right )$ and $\left ( 0,\infty \right )$ respectively. So in this case should the formula you suggested be changed in any way? – tzimhs panousis Mar 21 at 12:18
• @tzimhspanousis What I have written is still valid. When you calculate the inner integral just split it into $\int_{-\infty} ^{0}$ and $\int_0^{t/y}$ and use appropriate expressions for the density $f_X(u)$. – Kavi Rama Murthy Mar 21 at 12:21