A Monte Carlo Primer: A Practical Approach to Radiation by Stephen A. Dupree, Stanley K. Fraley

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By Stephen A. Dupree, Stanley K. Fraley

The mathematical means of Monte Carlo, as utilized to the shipping of sub-atomic debris, has been defined in different experiences and books considering its formal improvement within the Nineteen Forties. each one of these tutorial efforts were directed both on the mathematical foundation of the procedure or at its functional software as embodied within the numerous huge, formal desktop codes to be had for appearing Monte Carlo delivery calculations. This ebook makes an attempt to fill what seems to be a niche during this Monte Carlo literature among the math and the software program. therefore, whereas the mathematical foundation for Monte Carlo delivery is roofed in a few element, emphasis is put on the applying of the strategy to the answer of useful radiation delivery difficulties. this can be performed through the use of the computer because the easy instructing instrument. This ebook assumes the reader has a data of crucial calculus, neutron shipping concept, and Fortran programming. It additionally assumes the reader has to be had a laptop with a Fortran compiler. Any laptop of average measurement could be enough to breed the examples or resolve the workouts contained herein. The authors think it is crucial for the reader to execute those examples and routines, and by means of doing so that you can develop into comprehensive at getting ready acceptable software program for fixing radiation delivery difficulties utilizing Monte Carlo. The step from the software program defined during this publication to using creation Monte Carlo codes might be undemanding.

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The set of all B A stochastic event is one for which the outcome cannot be predicted with certainty. Games of chance such as the throwing of dice involve stochastic events. 21 22 Chapter 2 possible events accessible to the random variable is called the sample space for the random variable. The mathematical definition of a random variable is: Definition: A random variable V on a sample space U is a function from U into the set of real numbers R such that the inverse image of every interval in R is an event in U.

B The argument on the random variable V, defined on the sample space U, refers to the fact that the random variable V is taken on the sample space variable u in U. The probability of a random variable V being in the set {s} will be designated P{s}. Since U is a sample space, the probability of any event in U is well defined and, therefore, the probability P{a ~ V(u) ~ b} is well defined. Both the random variable and the sample space can be either continuous or discrete. As an example of a discrete random variable on a discrete sample space consider the throwing of a pair of dice, one of which is blue and the other red.

It is not unusual to encounter problems where the value of samples may differ in magnitude by factors of 108 over the range of interest. In such problems the standard deviation of the mean of the random variable will be much larger than that obtained for this simple integral, for the same number of samples and assuming the use of analog Monte Carlo. Accurate results from calculations of such a mean would be essentially impossible without some use of variance reduction. 26) -00 it can be seen that, if all of the samples from a population are equal to the mean value of the population, the variance is zero and each estimate is equal to the mean.

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