Bivariate and Trivariate Normal Distributions

Abstract

The following sections are included:IntroductionConicsQuadratic Expressions and Quadratic FormsThe General Expression of an EllipseSome ExamplesCorrespondence Between Coefficients When the Ellipse Center is Not at the OriginBivariate Normal DistributionNormal random vector V = (X, Y) ∼ N2(μ, Σ)Iso-density ellipsesStandard form of the binormalApplicationCharacteristic Function of the Bivariate NormalCdf of the Bivariate NormalDependence and Independence of ComponentsFrom dependent components to uncorrelated onesGoing from independent components to dependent onesGoing from dependent components to independent ones and backOrthant Probabilities for the Standard BinormalEllipses of Prediction (Dispersion or Prevision Ellipses)Ellipsoids and Their Projections into Lower Dimension SpacesChecking Bivariate NormalityTransformation to Reach Binormality and TrinormalityComputing Probabilities for the BinormalRecent Approaches to the Computation of the cdf of the Binormal and Trinormalcdf of the binormalcdf of the standard trinormalA Result of PlackettIndependence of Sample Mean and Sample VarianceDecomposition of an n-Sample of Binormal ObservationsA Distribution Directly Derived from the Binormal: The Bivariate Half-Normal DistributionDomains of Applications of the Binormal DistributionSpecial Bivariate Normal Distributions Used in Applied FieldsThe circular normal distributionPolar formTrivariate Normal DistributionThe standard trivariate normalThe general trivariate normalSpherical coordinatesBasic Properties of the Standard Trivariate NormalThe characteristic functionConditional distributionsOrthant Probability for the TrinormalTransformation of a Standard Trinormal Vector into a Trinormal Vector with Independent Componentscdf of the Standard TrinormalIso-probability Surfaces for a Standard TrinormalSpherical CaseSpecialized Tools for Computation of the Bivariate and Trivariate Normal DistributionsGeneration of Observations from a Bivariate and a Trivariate Normal DistributionBivariate caseTrivariate caseGeneral caseElicitation of a binormal distributionNormal CopulaMATLAB exampleMarginal distributionsConclusionComputational AppendixIntroductionPractical Volumes for Computing and Programming with the MultinormalUnivariate normalBivariate normalTrivariate normaln-variate normal (n ≥ 4)Critical and Prediction Ellipses and EllipsoidsPrediction ellipses (also called dispersion or prevision, or error ellipses), and critical ellipses and ellipsoids, in function of space dimensionsSome notations used in this bookPrediction ellipsoids in Rp of size α, for p ≥ 3Study of Cuts of the Normal Surface in R2 and R3Different kinds of cutsTransversal cuts (made at different heights, parallel to the horizontal plane)Practical problems related to cutsConclusionBibliography