For the exponential distribution, the cdf is .
For the exponential distribution, on the range of .
For the exponential distribution, the solution proceeds as follows.
In the case of exponential distribution
In practice, since both AND are uniformly distributed random
number, so the calculation can be simplified as
Because is equivelant to , and is a non-decreasing function (so that if then ) we get is equivelant to , which implies that which is equivelant to .
Once we have this procedure established, we can proceed to solve other similar distribution for which a inverse function is relatively easy to obtain and has a closed formula.