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 .