Stat 5101 Lecture Notes - School of Statistics

B.5. CONTINUOUS MULTIVARIATE DISTRIBUTIONS 223B.5 Continuous Multivariate DistributionsB.5.1The Uniform DistributionThe uniform distribution defined in Section B.2.1 actually made no mentionof dimension. If the set S on which the distribution is defined lies in R n , thenthis is a multivariate distribution.Conditional Distributions Every conditional distribution of a multivariateuniform distribution is uniform.Marginal Distributions No regularity. Depends on the particular distribution.Marginals of the uniform distribution on a rectangle with sides parallelto the coordinate axes are uniform. Marginals of the uniform distribution on adisk or triangle are not uniform.B.5.2The Standard Normal DistributionThe distribution of a random vector Z =(Z 1 ,...,Z k ) with the Z i i. i. d.standard normal.MomentsE(Z) =0var(Z) =I,where I denotes the k × k identity matrix.B.5.3The Multivariate Normal DistributionThe distribution of a random vector X = a + BZ, where Z is multivariatestandard normal.MomentsE(X) =µ=avar(X) =M=BB ′The Abbreviationcontext.N k (µ, M) orN(µ,M) if the dimension k is clear fromThe Sample Space If M is positive definite, the sample space is R k .Otherwise, X is concentrated on the intersection of hyperplanes determinedby null eigenvectors of MS = { x ∈ R k : z ′ x = z ′ µ whenever Mz =0}