= Probability density of a product of random variables
{title2=$f_{XY}(z)=\int_{\mathbb R}|x|^{-1}f_{X,Y}(x,z/x)\,dx$}
If $(X,Y)$ has a <joint probability density> $f$, the product has the displayed <probability density function>, interpreted almost everywhere. For fixed $x\ne0$, substitute $z=xy$ in the <integral> defining $\mathbb P(XY\in A)$; the absolute <Jacobian determinant> is $|x|$. The <Tonelli theorem> then exchanges the nonnegative <integrals>. The line $x=0$ has two-dimensional <Lebesgue measure> zero, so its omission creates no <probability mass>. A density may be infinite at exceptional points without creating an atom.
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