Let us first assume that the transformation is monotone increasing, which results in for all x.
Physically, the mathematical transformation must conserve probability, i.e., the probability of the r.v. occurring in dx about x must be the same as the probability of the r.v. occurring in dy about y, since if x occurs, the 1-to-1 relationship between x and y necessitates that y 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 .
Figure 11: Transformation of pdf's.
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 x, the fact that must be positive (by definition of probability) leads to the following expression for :