We often want to transform random variables and would like to get the same information about the transformed variable as the original.

  • Transforming to with . If we know the PDF of , what is the PDF of ? We can have linear transformations (e.g. Celsius to Fahrenheit) or nonlinear (dog to human age, apparently).
  • Transforming a random vector into and wanting to find the joint PDF of , given ‘s. An example is rectangular to polar coordinates.
  • Given that are independent r.v.s, what is the distribution of its sum or average?

Change of variables

One dimension

Suppose is a continuous r.v. with PDF . Let , where is differentiable and either strictly increasing or decreasing on .

If does not meet these requirements e.g. then we have to derive using the CDF of .

Then the PDF of is

Here, we’re taking the derivative of with respect to , where , and then the absolute value of that. Remember that , so we can calculate whichever one is easier.

  • The support of is given by all for .