Thus the use of elaborate simulation models to meet ideal conditions, rather than practical needs, is often foolish. Math. a= (9.3-6) and we finally have (9.3-7) ,.., C)(b) - c)(a) As a simple check on this formula let kl = 10 and k2 = n. then = -en + 1)/.jii. [335] J36] AN ESSAY ON SIMULATION EX8Dlple 10.8-3 CHAPTER 10 The Reciprocal Distribution To get mantissas for simulating floating point numbers from the reciprocal distribution we rely on the observation that succesive products from a flat distribution, (provided we shift off any leading digits that are 0, rapidly approaches the reciprocal distribution. Such simulations often reveal "what is going on" so that you can then approach the problem with the proper analytical tools and get solutions which are "understandable by the human mind" rather than get tables of numbers and pictures of curves. You are advised, however, because there are still a number of very bad generators in use, to make a few of your own tests that seem to be appropriate for the planned use before using it. That is, we use the simple formula, where the Xi are the random numbers from the uniform distribution and the Yi are those we want, (shift) with shifting to remove any leading zeros in the products. As above we get Solving for y we get y:.-ln(l-x) but since x is uniform from 0 to 1 we can replace 1 - x by x to get y = -lnx as the transformation. It is the same situation as in the use of non parametric or parametric statistics; nonparametric statistics gives poorer bounds than parametric statistics, but if the assumed model is seriously wrong then the parametric statistics result is almost surely worse! Still further thought suggests that the original distribution need not be finite in range, though again some limitations must be applied on the source distributions. To solve a number of problems in particle physics Fermi did a some simulations back in the late '30s. [339] J40] REFERENCES [Ke] Keynes, J. M. A Treatise on Probability, MacMillan, 1921 [Kh] Khinchine, A. I. It may, at times, be worth thinking of the detailed programming of the proposed simulation on some computer even if you have no intention of doing it-the act of programming, with its demands on explicitly describing all the details, often clarifies in your mind the muddled situation you started with. You can now compute the area by the standard formula 1 Xl Yl A= ~ 1 1 X2 Y2 X3 Y3 and take the absolute value of this result to get a positive area. Stat. Another aspect, greatly neglected by the experts, is the question of the believability of the simulation by those who have the power of decision. Sampling fluctuations, for reasonably sized samples, will not bother you much since you can often increase the size of the sample until it is very unlikely to be the cause of the difference. Theory of Probability, John Wiley and Sons, N.Y. 1970 [D] Diner, S., Fargue, D., Lochak, G., and Selleri, F., The Wave-Particle Dualism, D. Reidel Pub. Press, 1987 Chap 6. But we saw that if we regard the spikes as rectangles then we will get a nice normal approximation in the limit. We often need random samples from distributions other than the flat distribution which the random number generator supplies. In the late '20s the Bell Telephone laboratories ran simulations of the behavior of proposed central office switching machines using a combination of girls with desk calculators plus some simple mechanical gear and random number sources. The approximation of the binomial distribution by the normal is reasonable for moderate k. If we picture the original binomial coefficients as rectangles of width 1 centered about their values then we see that the approximating integral should run from 1/2 less than the lowest term to 1/2 above the highest term. If we have discrete distributions, say for example 6(x - 1/2) + 6(x 2 + 1/2) where d(x) is the usual delta function (a function with a single peak of no width but with total area 1), then the first convolution will give a distribution of 1/4, 1/2, 1/4 and the following ones will generate the corresponding binomial coefficients. Exercises 10.3 Simulate finding the area of the triangle in Example 10.3-1. By trying the same sequence of calls on the alternate designs we could evaluate which design would have done better (in terms of the measure of success we have decided on) for that particular sequence of calls.