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 :