TITLE: What exactly is the Probability Integral Transform?
QUESTION [15 upvotes]: I've been going back over my notes from Stats class and came across the Probability Integral Transform.  From my limited understanding, the basic idea is that a cdf in terms of one variable can be transformed into another cdf in terms of different variable:

i.e. from $F_x(x)$ to --> $F_y(y)$

Is this understanding correct?  What is the purpose behind this?  Finally, is there a general procedure in performing the transformation?

REPLY [14 votes]: Your understanding looks basically correct to me.
As far as purpose, I've seen it used mostly to generate random variables from continuous distributions.  For instance, if $X$ has a $U(0,1)$ distribution, then $F_X(x) = x$.  Thus the requirement $F_X(x) = F_Y(y)$ in the probability integral transform reduces to $x = F_Y(y)$ or $y = F_Y^{-1}(x)$.  Since $y$ is an observation from the probability distribution $Y$, this means that we can generate observations from the distribution $Y$ by generating $U(0,1)$ random variables (which most software programs can do easily) and applying the $F_Y^{-1}$ transformation.  
For example, suppose you want to generate instances of an exponential$(\lambda)$ random variable.  The cdf is $$F(y) = \int_0^y \lambda e^{-\lambda t} dt = 1 - e^{-\lambda y}.$$
Solving for $y$, we have $$F(y) - 1 = - e^{-\lambda y} \Rightarrow -\lambda y = \ln (1- F(y)) \Rightarrow y = F^{-1}(x) = -\ln(1-x)/\lambda.$$
Thus if $x$ is an observation from a $U(0,1)$ distribution, then $y = -\ln(1-x)/\lambda$ is an observation from an exponential$(\lambda)$ distribution.  Moreover, $x$ having a $U(0,1)$ distribution is equivalent to $1-x$ having a $U(0,1)$ distribution, so we often express the transformation as $y = -\ln x/\lambda$.
As far as a general procedure for performing the transformation, what I've done here with the uniform and exponential distributions should give you a guide.  Unfortunately, though, there aren't that many commonly-used distributions for which the cdf can be inverted analytically.