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Given two random variables $x, y$, each with the probability distribution functions $p_x(x)$, $p_y(y)$, then if $z = xy$, then $p_z(z) = \int p_x(x)p_y(z/x)\frac{1}{|x|}dx $.

Is there a similar proof that gives the cumulative distribution for $z$ given the cumulative distribution functions for $x$ and $y$?

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    $\begingroup$ Ordinarily, a "probability distribution function" is another term for CDF. Evidently you are referring to probability density functions. $\endgroup$
    – whuber
    Aug 21, 2017 at 20:46

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