You're correct in that this allows you to simulate random variables of different distributions. It may help to explain with an example.
Suppose I'm simulating a two-parmeter pareto distribution, with a cumulative distribution function of F(X) = 1 - [ B / (B + X) ] ^ A, where A = 2 and B = 5.
Here's how the cumulative distribution function of this pareto distributionn would look for a few different values of X:
All the above does is replace X in the CDF with the numbers 1-15 and solve the CDF equation with A=2 and B=5.
Now we want to simulate a pareto value using the uniform distribution. So we ask for a random, uniform number between 0 and 1 and let's suppose that we get 0.872. Based on the above, the random number from the pareto distribution that corresponds to the uniform random number 0.872 is 9, since the value of 9 corresponds to the 87.2% point on the pareto cumulative distribution function.
So in this case, X is the random number generated by the uniform random number generator (0.872) and F^-1(U) is the pareto value that has a cumulative distribution function value corresponding to it (9).
Does that help?
- Steve J