By Stephen A. Dupree, Stanley K. Fraley

The mathematical means of Monte Carlo, as utilized to the delivery of sub-atomic debris, has been defined in several reviews and books on account that its formal improvement within the Forties. every one of these tutorial efforts were directed both on the mathematical foundation of the method or at its functional program as embodied within the a number of huge, formal machine codes to be had for acting Monte Carlo delivery calculations. This booklet makes an attempt to fill what seems to be a spot during this Monte Carlo literature among the math and the software program. therefore, whereas the mathematical foundation for Monte Carlo shipping is roofed in a few element, emphasis is put on the applying of the strategy to the answer of useful radiation delivery difficulties. this is often performed through the use of the computer because the easy instructing software. This publication assumes the reader has an information of crucial calculus, neutron shipping thought, and Fortran programming. It additionally assumes the reader has to be had a computer with a Fortran compiler. Any computer of moderate measurement will be enough to breed the examples or clear up the workouts contained herein. The authors think it will be significant for the reader to execute those examples and workouts, and by way of doing in an effort to turn into comprehensive at getting ready acceptable software program for fixing radiation shipping difficulties utilizing Monte Carlo. The step from the software program defined during this e-book to using construction Monte Carlo codes may be effortless.

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A, 124, 1961, pp. 227-239. 2 A. Hall, "On an Experimental Determination of1t," Messeng. Math. 2, 1873, pp. 113·14. I 1. Introduction 19 ) Lord Kelvin, "Ninteenth Century Clouds over the Dynamical Theory of Heat and Light," Phil Mag 6,2, 1901, pp. 1-40. 4 W. S. Gosset, "Probable Error ofa Correlation Coefficient," Biometrika 6, 1908, p. 302. s A. S. Householder, G. E. Forsythe, and H. H. S. S. , 1951. 6 The Rand Corporation, A Million Random Digits with 100,000 Normal Deviates, Free Press Publishers, Glencoe, IL, 1955.

26) -00 Again, the standard deviation cr of a continuous random variable V is the positive square root of the variance of V. 29 is useful for calculating or estimating the variance of a random variable. It is valid for both discrete and continuous random variables. However, the expected value of a random variable may not exist, or may not be finite. If the expected value is not finite, the variance does not exist. If the expected value is finite, the variance exists but may not be finite. We will see examples of the latter in Chapter 7.

7,' stdevof sum:',eI4 . 6) REIURN END For many random variables of interest, the range over which a Monte Carlo estimate varies within a stratum can be less than that for the total sample space. Since the variance of the variable depends upon this range, and reducing the range reduces the variance, stratification generally reduces the variance. In fact, provided the sampling is distributed pro rata among the strata (the number of samples in each stratum is proportional to the "size" of the stratum), in no case will stratified sampling reduce the accuracy of a Monte Carlo estimate.