Suppose that we are given an random variable X that is uniformly distributed over the open interval (0,1) and that we wish to generate sample values for another random variable Y with probability distribution function F₁(y). To do so, we find a monotonically increasing function y=g(x) and then simply evaluate it at sample values from the random variable X. The function y=g(x) is defined by the inverse of F₁(y) restricted to the domain (0,1).

Example.

Suppose that we want to generate values from an exponential pdf F₁(y)=1-exp(-a*y) if y is positive, and F₁(y)=0 if y non-positive. Since F₁^-1(y) = -ln(1-y)/a, the function of interest is y=g(x)=-ln(1-x)/a.