Random Numbers part 3

In the previous section, we learned how to generate random deviates with a uniform probability distribution, so that the probability of generating a number between x and x + dx, denoted p(x)dx, is given by p(x)dx = dx 0
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Random Numbers part 3 7.2 Transformation Method: Exponential and Normal Deviates 2877.2 Transformation Method: Exponential and Normal Deviates In the previous section, we learned how to generate random deviates witha uniform probability distribution, so that the probability of generating a number visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)between x and x + dx, denoted p(x)dx, is given by dx 0 < x < 1 p(x)dx = (7.2.1) 0 otherwiseThe probability distribution p(x) is of course normalized, so that ∞ p(x)dx = 1 (7.2.2) −∞ Now suppose that we generate a uniform deviate x and then take some prescribedfunction of it, y(x). The probability distribution of y, denoted p(y)dy, is determinedby the fundamental transformation law of probabilities, which is simply |p(y)dy| = |p(x)dx| (7.2.3)or dx p(y) = p(x) (7.2.4) dyExponential Deviates As an example, suppose that y(x) ≡ − ln(x), and that p(x) is as given byequation (7.2.1) for a uniform deviate. Then dx p(y)dy = dy = e−y dy (7.2.5) dywhich is distributed exponentially. This exponential distribution occurs frequentlyin real problems, usually as the distribution of waiting times between independentPoisson-random events, for example the radioactive decay of nuclei. You can alsoeasily see (from 7.2.4) that the quantity y/λ has the probability distribution λe−λy . So we have#include float expdev(long *idum)Returns an exponentially distributed, positive, random deviate of unit mean, usingran1(idum) as the source of uniform deviates.{ float ran1(long *idum); float dum; do dum=ran1(idum); while (dum == 0.0); return -log(dum);}288 Chapter 7. Random Numbers 1 uniform ⌠ y deviate in F(y) =⌡ 0 p(y)dy x p(y) visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108 ...