som e Generating Continuous Random Variables Quasi random numbers So far we learned about pseudo random sequences and a common method for generating them These will be the inputs to our Monte Carlo processes Alternate source Quasi random numbers They are not really random so they have to be used very carefully They are not used very much for Monte Carlo methods in radiation transport But because they are of such importance to the general field of Monte Carlo i e beyond transport methods you should have a taste of what they are how to get and use them Quasi random numbers 2 Quasi random numbers are not random at all they are a deterministic equal division sequence that are ordered in such a way that they can be used by Monte Carlo methods The easiest to generate are Halton sequences which are found by reflecting the digits of prime base counting integers about their radix point You pick a prime number base Count in that base Reflect the number Interpret as base 10 Quasi random numbers 3 After M BaseN 1 numbers in the sequence the sequence consists of the equally spaced fractions i M 1 i 1 2 3 M Therefore the sequence is actually deterministic not stochastic Like a numerical method The beauty of the sequence is that the re ordering results in the an even coverage of the domain 0 1 even if you stop in the middle Because you need ALL of the numbers in the sequence to cover the 0 1 range which is not true of the pseudo random sequence it is important that all of the numbers of the sequence be used on the SAME decision For a Monte Carlo process that has more than one decision a different sequence must be used for each decision This is why we don t use them The most common way this situation is handled is to use prime numbers in order 2 5 7 11 etc for decisions Halton sequence after UNC log10 N d 1 professor N Asymptotic estimate of error is which means closer to 1 N The resulting standard deviations calculated and printed by standard MC codes are NOT accurate Get estimates by oversetting som e Generating Continuous Random Variables Generating Continuous Random Variables Suppose that X is a continuous random variable with cdf F x Furthermore suppose that you can invert F Note that if U unif 0 1 the random variable has the same distribution as X F 1 U Generating Continuous Random Variables Example To generate exponential rate rv s x f x e x 0 pdf x cdf F x e u du 1 e x x 0 0 1 F x ln 1 x 1 Now plug in a uniform Generating Continuous Random Variables The standard normal distribution 1 x2 2 1 f x e 2 x There is no closed form expression for the cdf normal Generating Continuous Random Variables The Box Muller Transformation Let U1 and U2 be independent unif 0 1 rv s Then X1 2 ln U1 cos 2 U2 is normally distributed with mean 0 and variance 1 So is X2 2 ln U1 sin 2 U2 and X1 and X2 are independent normal Generating Continuous Random Variables Acceptance Rejection Method want to simulate a rv X with pdf f need to find another function g so that g x f x x 1 normalize g to a pdfh x g x c where c g x dx want to simulate a rv X with pdf f need to find another function g so that g x f x x 1 normalize g to a pdfh x g x c where c g x dx generate Y with pdf h generate U unif 0 1 indep of Y if U f Y g Y accept Y set X Y otherwise reject Y return to Proof discrete case P X x P X x andfirstaccepthappenson nth try n 1 f y1 f yn 1 f x P Y1 y1 U1 Yn 1 yn 1 Un 1 Yn x Un g y1 g yn 1 g x n 1 f x P no x s P Yn x Un g x n 1 f x P no x s P Y x U g x n 1 Proof continued f x P no x s P Y x U g x n 1 f x no x s P Y x U g x f x g x no x s g x f x no x s Proof continued So using this algorithm P X x f x no x s Sum both sides over x P X x no x s f x x x 1 no x s 1 no x s 1 What we wanted P X x f x Acceptance Rejection Method Example Example Target Density f x 9x e 3x x 0 2x 1 f2 x 2e g x 9e e 2x Try g x ce 2x c 9e 1 works Acceptance Rejection Method Example RESULTS 100 000 draws Target Density f x 9x e 3x x 0