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3.2.3 Inverse Cumulative Distribution Function

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.)