Combining the two cases leads to the following simple rule for transforming pdf's:
For multidimensional pdf's, the derivative is replaced by the Jacobian of the transformation, which will be described later when we discuss sampling from the Gaussian pdf.
Consider the elastic scattering of neutrons of energy from a nucleus of mass A (measured in neutron masses) at rest. Define as the probability that the final energy of the scattered neutron is in the energy interval dE about E, given that its initial energy was . The pdf is given by:
We now ask: what is the probability that the neutron scatters in the speed interval dv about v, where ? Using Eq. (79), one readily finds the following expression for the pdf :
It is easy to show that is a properly normalized pdf in accordance with Eq. (24).