Probability Distributions

1. Continuous Distributions1.1 Introduction1.2 Sample Data Statistics1.3 Notation1.4 Parameter Estimating Methods1.5 Transforming VariablesTransform Data to (0,1)Transform Data to (x â ¥0)1.6 Continuous Random Variables1.7 Continuous Uniform Coefficient of VariationParameter Estimates1.8 Exponential 1.9 Erlang Parameter Estimates1.10 Gamma Parameter Estimates1.11 Beta Standard Beta Mean and Variance Parameter Estimates1.12 Weibull Weibull Plot Parameter Estimates1.13 NormalStandard Normal Distribution Coefficient of VariationParameter Estimates1.14 Lognormal Parameter Estimates1.15 Summary1.16 Reference2 Discrete Distributions2.1 Introduction2.2 Discrete Random VariablesLexis Ratio2.3 Discrete UniformParameter Estimates2.4 BinomialLexis RatioParameter EstimatesNormal Approximation Poisson Approximation 2.5 GeometricNumber of TrialsNumber of FailuresLexis RatioParameter Estimate2.6 PascalNumber of TrialsLexis Ratio Parameter EstimateNumber of FailuresLexis RatioParameter Estimate2.7 PoissonLexis RatioRelation to the Exponential DistributionParameter Estimate2.8 Hyper GeometricParameter Estimate2.9 Summary2.10 Reference3 Standard Normal3.1 Introduction3.2 Gaussian Distribution3.3 Some Relations on the Standard Normal Distribution4.3 Normal Distribution3.5 Standard Normal3.6 Hastings ApproximationsApproximation of F(z) from zApproximation of z from F(z)3.7 Table Values of the Standard Normal3.8 Discrete Normal Distribution3.9 Summary3.10 References4 Partial Expectation4.1 Introduction4.2 Partial Expectation4.3 Left Location ParameterTable Entries4.4 Inventory Management4.5 Right Location Parameter4.6 Advance Demand4.7 Summary4.8 References5 Left Truncated Normal5.1 Introduction5.2 Left-Location Parameter5.3 Mathematical Equations5.4 Table Entries5.5 More Tables5.6 Left Truncated Distribution5.7 Application to Sample Data5.8 LTN for Inventory ControlAutomotive Service Parts Distribution CenterRetail Products5.9 Summary5.10 References6 Right Truncated Normal6.1 Introduction6.2 Right Truncated Distribution6.3 Mathematical Equations6.4 Variable t Range6.5 Table Entries6.6 Application to Sample Data6.7 More Tables6.8 Summary6.9 Reference7 Truncated Normal Spread Ratio7.1 Introduction7.2 The Spread Ratio7.3 LTN Distribution Measures7.4 LTN Table Entries7.5 RTN Distribution Measures7.6 RTN Table Entries7.7 Estimating the Distribution Type7.8 Selecting the Distribution Type7.9 Estimating the Low and High LimitsWhen LTNEstimate ï § When LTNWhen RTNEstimate ï ¤When RTNWhen NormalCompute the Adjusted Coefficient of Variation7.10 Find xï ¡ï  where P(x â ¤ xï ¡) = ï ¡7.11 Find ï ¡ where P(x â ¤ x`) = ï ¡7.12 Summary 8 Bivariate Normal 8.1 Introduction 8.2 Bivariate Normal DistributionMarginal DistributionsConditional Distributions8.3 Bivariate Standard Normal DistributionConditional Distribution of z2Conditional Distribution of z1Cumulative Joint ProbabilityApproximation of F(k1,k2)Table Values of F(k1,k2) 8.4 Some Basic Probabilities for (z1,z2) ~ BVN(0,0,1,1,ï ²)8.5 Probabilities for (x1,x2) ~ BVNï ¨ï ­ï ±ï ¬ï ­ï ²ï ¬ï  ï ³ï ±ï ¬ï ³ï ²ï ¬ï ²ï ©8.6 Summary8.7 References9 Lognormal9.1 Introduction9.2 Lognormal Distribution9.3 Notation9.4 LognormalLognormal Mode Lognormal Median9.5 Raw Lognormal Variable9.6 Shifted Lognormal Variable9.7 Normal Variable9.8 Zero-Mean Normal Variable9.9 Standard LN Variable9.10 Lognormal Table Entries9.11 Lognormal Distribution Table9.12 Summary9.13 Reference10 Bivariate Lognormal 10.1 Introduction10.2 Bivariate Lognormal NotationSome Properties Between x and yMode of x and x`10.3 Lognormal and Normal NotationRelated Parameters10.4 Bivariate Lognormal DistributionBivariate Lognormal CorrelationBivariate Lognormal Designation10.5 Bivariate Normal Distribution10.6 Bivariate (Zero-Mean) Normal Distribution10.7 Bivariate (Standard) Normal Distribution10.8 Summary10.9 References