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.