In order to have a complete discussion of sampling, we need to explain transformation rules for pdf's. That is, given a pdf , one defines a new variable , and the goal is to find the pdf that describes the probability that the r.v. y occurs. For example, given the pdf for the energy of the scattered neutron in an elastic scattering reaction from a nucleus of mass A, what is the pdf for the speed v, where ?
First of all, we need to restrict the transformation to be a unique transformation, because there must be a 1-to-1 relationship between x and y in order to be able to state that a given value of x corresponds unambiguously to a value of y. Given that is 1-to-1, then it must either be monotone increasing or monotone decreasing, since any other behavior would result in a multiple-valued function .