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Monte Carlo Process Charles Yoe, Ph.D. College of Notre Dame of Maryland

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Analytical Solution lSolution meets all criteria of problem 2 + x = 4 x = 2 is a solution that works lSome problems have more than one analytical solution x 2 = 9 lSome problems have no analytical solution

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Simulation lNumerical technique used to estimate analytical solutions to a problem lNot an optimization technique, answers what-if questions lResults are not analytical solutions lAnalytical solutions are preferred

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Monte Carlo Process lCode name for simulations relating to development of atomic bomb lApplied to wide variety of complex problems involving random behavior lProcedure that generates values of a random variable based on one or more probability distributions lNot simulation method per se

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Suppose... lYou have a variable that varies between 10 and 50 lAll you know is theoretical maximum and minimum, any number between is equally likely

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Monte Carlo Process lIs a process that can generate numbers within that range lAccording to the rules you specify In this case a min and a max Any number as likely as any other number

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Monte Carlo Process lTwo steps lGenerate a simple random number lTransform it into a useful value using a specific probability distribution

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Random Number Generation lPseudorandom Numbers [0,1] lSeed = 6721 (any number) lMid-square Method (John von Neumann) (6721) 2 = ; r1= (1718) 2 = ; r2= (5152) 2 = ; r3= etc. lMore sophisticated methods now used

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Transformation (1) lAssume Uniform Distribution, U(a,b) where a = 10 and b = 50 lTo obtain a value, x, we use x = a + (b - a)u lIn this case, x = u

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Transformation (2) lGenerate U~U(0,1), say u = then x = 10 +( ) = 16.9 x = 10 +( ) = 30.6 x = 10 +( ) = 31.7, etc. lOther distributions are similar but more complex transformations

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Some Language lIteration--one recalculation of the model during a simulation. Uncertain variables are sampled once during each iteration according to their probability distributions lSimulation--technique for calculating a model output value many times with different input values. Purpose is to get complete range of all possible scenarios

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Monte Carlo Simulation lSimulation model that uses the Monte Carlo process lDeterministic values in models replaced by distributions lValues randomly generated for each probabilistic variable in model and calculations are completed lProcess repeated desired # times

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Monte Carlo Simulation X =

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How Many Iterations? lMeans often stabilize quickly--few hundred lEstimating probabilities of outcomes takes more lDefining tails of output distribution takes many more iterations lIf extreme events are important it make take many many more

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Some Examples lThe Monte Carlo process is used for several risk assessments linked to the Clearinghouse Salmonella Antimicrobial Resistant Campylobacter

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The End

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