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.