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 
.
