The eminent mathematician John von Neumann devised an efficient and elegant method to circumvent the rejections. This consists of formulating the cumulative distribution function (CDF) for emission, as follows:
The CDF is the cumulative probability that an emission has occurred between and . The range of a CDF is between 0 and 1. The advantage of the CDF is that, by aggregating all emissions into a cumulative distribution, an emission is obtained in every trial (we are certain that a particle will be emitted in some direction from the surface)---thus, no work is wasted.
We proceed by replacing the CDF with a random number, , and invert the function:
yielding the inverse CDF shown in Figure 10. Note that, a continuum of uniformly distributed random numbers yields the distribution of the argument in the integral, i.e.,
(this is apparent because a uniform distribution of random numbers places equal numbers into uniformly spaced bins, as depicted graphically in Figure 10, where emissions are concentrated about ). In fact, differentiating equation (20) yields , substantiating this argument.
Figure 10: Inverse CDF for Emission.
Thus, the final algorithm for emission is as follows:
(Note that, here we have dropped the hat on .)
(See exercise 7.)