 Change of variables for probability densities

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.