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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 $Z$?

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

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  • $\begingroup$ 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? $\endgroup$ – tzimhs panousis Mar 21 at 12:18
  • $\begingroup$ @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 $f_X(u)$. $\endgroup$ – Kavi Rama Murthy Mar 21 at 12:21

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