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LECTURE 25 SIMULATION AND MODELING Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411

# LECTURE 25 SIMULATION AND MODELING Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411

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• LECTURE 25SIMULATION AND MODELINGMd. Tanvir Al Amin, Lecturer, Dept. of CSE, BUETCSE 411

• Recap : Inverse Transform MethodFind cumulative distribution functionLet it be FFind F-1X = F-1(R) is the desired variate where R ~ U(0,1)Other methods areAccept Reject techniqueComposition methodConvolution techniqueSpecial observations

• Random variate for Continuous DistributionWe already covered for continuous case :Proof of inverse transform method(Kelton 8.2.1 pg440)Exponential distribution (Banks 8.1.1)Uniform distribution (Banks 8.1.2)Weibull distibution (Banks 8.1.3)Triangular distribution (Banks 8.1.4)Empirical continuous distribution (Banks 8.1.5)Continuous distributions without a closed form inverse (Self study Banks 8.1.6)

• Banks Chapter 8.1.7 + Kelton 8.2.1Inverse Transform method for Discrete Distributions

• Things to rememberIf R ~ U(0,1), A Random variable X will assume value x whenever,

Also we should know the identities :

We could use this identity also, It results in F-1(R) = floor(x) which is not useful because x is already an integer for discrete distribution.

• Discrete DistributionsR1 = 0.65F(x) = F(1) = 0.8F(x-1) = F(0) = 0.4X = 1

• Empirical Discrete DistributionSay we want to find random variate for the following discrete distribution.

At first find the F

xp(x)00.5010.3020.20

xp(x)F(x)00.500.5010.300.8020.201.00

• Empirical Discrete Distribution

• Discrete Uniform DistributionConsider

• Discrete Uniform DistributionGenerate R~U(0,1)123K-1k , output X= 1 output X= 2 output X= 3, output X= i output X= 4If generated

• Discrete Uniform DistributionImp : Expand this method for a general discrete uniform distribution DU(a,b)

• Discrete Uniform DistributionAlgorithm to generate random variate for p(x)=1/k where x = 1, 2, 3, k

Generate R~U(0,1) uniform random numberReturn ceil(R*k)

• Geometric Distribution

Recall :

• Geometric DistributionAlgorithm to generate random variate for

Generate R~U(0,1) uniform random numberReturn ceil(ln(1-R)/ln(1-p)-1)

• Proof of inverse transform method for Discrete caseKelton 8.2.1 Page 444We need to show that

• Disadvantage of Inverse Transform MethodKelton page 445-448DisadvantageNeed to evaluate F-1. We may not have a closed form. In that case numerical methods necessary. Might be hard to find stopping conditions.For a given distribution, Inverse transform method may not be the fastest way

• Advantage ofInverse Transform MethodStraightforward method, easy to useVariance reduction techniques has an advantage if inverse transform method is usedFacilitates generation of order statistics.

• Kelton 8.2.2Composition Technique

• Composition TechniqueApplies when the distribution function F from which we wish to generate can be expressed as a convex combination of other distributions F1, F2, F3, i.e., for all x, F(x) can be written as

• Composition AlgorithmTo generate random variate forStep 1 : Generate a positive random integer j such that

Step 2 : Return X with distribution function Fj

• Geometric InterpretationFor a continuous random variable X with density f, we might be able to divide the area under f into regions of areas p1,p2,(corresponding to the decomposition of f into its convex combination representation)Then we can think of step 1 as choosing a regionStep 2 as generating from the distribution corresponding to the chosen region

• Example

0 a 1-a 1

• Example

• Proof of composition techniqueKelton page 449

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