To find the actual normal approximation parameters, as stated above we match the first three moments. We must acknowledge, however, that in practice, in corporations, in science, in engineering, and in industry generally, this mistake is repeatedly made; a lot of different sources are combined and then the rigorously found bounds of uncertainty on many of the parts are taken to also cover the parts that are wild guesses at best. 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. But when you are near an optimum then most variations of the design will be poorer than the current one, and due to the chance fluctuations of the simulation (the actual sample you used) on a single step of the optimization you would often choose an inferior design over a better one. To get a random point in the circle you pick a pair of random numbers in the range 0 to 1 and transform them to the range -1 to +1 by the simple transformation x' == -1 + 2x You then calculate the sum of the squares to find if the point is inside the unit circle. and Tribus, Myron, The Maximal Entropy Formalism, MIT Press, 1979 [M] Miller, R. W. Fact and Method, Princeton Uni. In table 9.4-1 we saw that even the sum of 3 random numbers from the flat random number generator closely approximated the normal distribution; the sum of 12 naturally does much better. Zipf tried to base it on the natural economy of the situation, following the then popular least action, least time, and least work laws that were of great importance in the growing field of mechanics. Co. 1984 [Ed] Edwards, A. W. F., Likelihood, Cambridge Uni. After years of effort on the problem there is still no completely satisfactory random number generator, and unless you are willing to devote a very large amount of effort you will probably settle for the generator supplied. We see from this Table that the normal distribution, even for only three random numbers from a flat distribution, is a remarkably good approximation. Math. 1989, p. 48-69. The New Positivism, Chap. 10.5 The Use of Some Modeling It may be that from other sources you have reason to believe that the interarrivals times of the calls are closely modeled by an appropriate exponential distribution. (The latter may well be true!) Indeed, Zipf's book is filled with many, quite diverse examples of this rule. In many respects it resembles the earlier claim that enough single events when added approach the normal distribution. With modern computers this is quite reasonable to try. In both cases it is indicative, and we rarely have reality so nice as to know the details, or how many independent sources there are. Example 10.8-1 SOME SIMPLE DISTRIBUTIONS Tbe Exponential Distribution Suppose we want random numbers from the distribution f(y) = exp( -V). of America [Ka] Kaplin, Mark. 10.3 When You Cannot Compute the Result Often you cannot analytically compute the results you want from the model, either because the mathematics at your command are inadequate, or because you simply cannot formulate the problem in a sufficiently mathematical form to handle it.