TheinversetransformmethodcanbeusedSupposethecumulativedistributionfunctionis:Fx (x )The steps to generate x; are:(1) generate a u; which is uniformly distributedin [O, 1] using a random number generatordiscussedintheabove section;X; = Fx'(u.)(2)11
The inverse transform method can be used. Suppose the cumulative distribution function is: FX xi The steps to generate xi are: (1) generate a ui which is uniformly distributed in [0, 1] using a random number generator discussed in the above section; (2) i 1 xi F u X 11
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Generation.of randomnumberswitha standard normal distributionThe steps are:(1)generate a pair of independent uniformly distributedrandom numbers, u, and uz in [0, 1];(2) Calculate s, and s, by the formulaeSt = /-2ln(u,)sin2元u2S2 = /-2ln(u,)cos 2元u213
Generation of random numbers with a standard normal distribution The steps are: (1)generate a pair of independent uniformly distributed random numbers, u1 and u2 in [0, 1]; (2) Calculate s1 and s2 by the formulae 1 1 2 u2 s 2ln u sin 2 1 2 u2 s 2ln u cos 13
Generationofrandomnumberswithlog-normal distributionThe steps are:(1)generate a random number with normal distribution,say n;(2)L; = eniSo L obeys log-normal distribution14
Generation of random numbers with log-normal distribution The steps are: (1)generate a random number with normal distribution, say ni ; (2) So Li obeys log-normal distribution i n i L e 14
CONTENTSMONTECARLOSIMULATION-GenerationofRandomNumbers-Generationofrandomnumberswithagiventypeofdistribution- Generationofrandomnumberswithastandardnormal distribution-Generationofrandomnumberswithlog-normaldistribution-ErrorEstimationofMonteCarloSimulationOTHERSIMULATIONBASEDMETHODSImportanceSamplingMethod-Selectionof ImportanceSamplingFunctionEXAMPLESCLOSINGREMARKS15
CONTENTS • MONTE CARLO SIMULATION – Generation of Random Numbers – Generation of random numbers with a given type of distribution – Generation of random numbers with a standard normal distribution – Generation of random numbers with log-normal distribution – Error Estimation of Monte Carlo Simulation • OTHER SIMULATION BASED METHODS – Importance Sampling Method – Selection of Importance Sampling Function • EXAMPLES • CLOSING REMARKS 15