Change of variables for probability densities ($f_Y(y)=f_X(h^{-1}(y))\left|\frac{d{h}^{-1}(y)}{dy}\right|$)

For a random variable $X$ with probability density function $f_X$ on an interval, let $Y=h(X)$, where $h$ is a diffeomorphism onto another interval. The change of variables formula givesIndeed, integrating the right-hand side over an outcome interval gives the probability that $X$ lies in its inverse image. The absolute derivative preserves positive probability when $h$ reverses orientation. A gravity deflection of an atomic beam is a physical example: its speed-to-height transformation changes the probability density and its mode.

If $(X,Y)$ has a joint probability density $f$, the product has the displayed probability density function, interpreted almost everywhere. For fixed $x\ne 0$, 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.