Lecture Notes

Functions of Random Variables

By Akshay Agrawal. Last updated Feb. 6, 2019.

Let be a random variable with support in and density , and let , where is a one-to-one, continuously differentiable mapping such that its Jacobian determinant is nonzero for all . Then the density of , which we denote , is given by

or equivalently

We can sketch a proof of this result by appealing to the change of variables theorem. For all subsets in the support of ,

On the other hand, under the change of variables , the integral in (2) is equal to

Since this is true for all subsets , we can conclude that the integrands in (1) and (3) are equal, giving

as desired.