Essential Cheat Sheet for Simulation Final Exam Preparation

School

Georgia Institute Of Technology**We aren't endorsed by this school

Course

ISYE 6644

Subject

Statistics

Date

Dec 11, 2024

Pages

Uploaded by BrigadierLeopardPerson1679

Simulation Final Exam Cheat Sheet1. Key Probability DistributionsUniform Distribution: All outcomes equally likely.oPDF: oMean: , Variance: Normal Distribution: Bell-shaped curve.oPDF: oMean: , Variance: Exponential Distribution: Time between events in Poisson process.oPDF: oMean: , Variance: Bernoulli: One trial.oPMF: oMean: , Variance: Binomial: Sum of Bernoulli trials.oPMF: oMean: , Variance: 2. Random Number GenerationUniform (0,1):Base for generating random numbers.oMethod: Linear Congruential Generator Inverse Transform Method: , where .oExample (Exponential): , so .Acceptance-Rejection Method: For generating from complicated PDFs.oSteps:1.Choose , a simpler PDF.2.Generate and .3.Accept if , where is a constant.3. Simulation BasicsMonte Carlo Simulation:oUse randomness to solve deterministic problems (e.g., integration, estimating probabilities).oSteps:1.Define domain of inputs.2.Generate random inputs.

3.Compute outputs.4.Aggregate results (e.g., mean, variance).Law of Large Numbers:oAs , the sample mean converges to the population mean .Central Limit Theorem:oFor large , .4. Statistical Output AnalysisPoint Estimation: Use sample statistics (e.g., sample mean, variance) to estimate population parameters.Confidence Intervals (CI):o, where is the sample standard deviation and is from the standard normal table.oCI: .Hypothesis Testing:oNull Hypothesis : Assumes no effect or difference.oTest Statistic (e.g., -test, -test): Compare observed data to .oP-value: Probability of observing test statistic as extreme as current data under .5. Queueing Theory BasicsKendall’s Notation: o: Arrival distribution (e.g., M for Poisson).o: Service time distribution (e.g., M for exponential).o: Number of servers.Key Metrics:o: Average number in system.o: Average number in queue.o: Average time in system.o: Average time in queue.oLittle’s Law: , .6. Variance Reduction TechniquesCommon Methods:oAntithetic Variates: Use negatively correlated variables.oControl Variates: Use known expected values of related variables.oStratified Sampling: Divide domain into strata and sample each.

7. Verification & ValidationVerification: Ensuring the model is implemented correctly (debugging).Validation: Ensuring the model represents the real-world system accurately.8. Software TipsPython Libraries:onumpy.randomfor generating random variables.oscipy.statsfor PDFs, CDFs, and statistical tests.R Functions:orunif(), rnorm(), rexp(), etc., for random variates.ot.test(), var.test()for hypothesis testing.