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 
:
