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. 
occurs.
For example, given the pdf 
for the energy of the
scattered neutron in an elastic scattering reaction from a nucleus of mass 
,
what is the pdf 
for the speed 
, 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 
and
in order to be able to state that a given value of corresponds unambiguously
to a value of . 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 .
Let us first assume that the transformation is monotone increasing,
which results in 
for all . Physically, the
mathematical transformation must conserve probability, i.e., the probability
of the r.v. 
occurring in 
about must be the same as the
probability of the
r.v. 
occurring in 
about , since if occurs, the 1-to-1
relationship
between and necessitates that appears.
But by definition of the pdf's and ,
The physical transformation implies that these probabilities must be equal.
Figure 11
illustrates this for an example transformation .
Equality of these differential probabilities yields
and one can then solve for :
This holds for the monotone increasing function . It is easy to show that
for a monotone decreasing function , where 
for all , the
fact that must be positive (by definition of probability) leads to the
following expression for :
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 (measured in neutron masses) at rest.
Define 
as the probability that
the final energy of the scattered neutron is in the energy interval 
about
, 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 
about , 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).
Perhaps the most important transformation occurs when is the cumulative
distribution function, or cdf:
In this case, we have 
, and one finds the important result that
the pdf for the transformation is given by:
In other words,
the cdf is always uniformly distributed on [0,1],
independently of the pdf !
Any value for the cdf is equally likely on the interval [0,1].
As will be seen next, this result has important ramifications for
sampling from an arbitrary pdf.
