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.