Stat 5101 Lecture Notes - School of Statistics

224 Stat 5101 (Geyer) Course NotesThe Parameters The mean vector µ and variance matrix M.The Density Only exists if the distribution is nondegenerate (M is positivedefinite). Thenf X (x) =1(2π) n/2 det(M) 1/2 exp ( − 1 2 (x − µ)′ M −1 (x − µ) ) ,x ∈ R kMarginal DistributionsAll are normal. If( )X1X =X 2is a partitioned random vector with (partitioned) mean vector)µ1E(X) =µ=(µ 2and (partitioned) variance matrixvar(X) =M=( )M11 M 12M 21 M 22and X ∼N(µ,M), thenX 1 ∼N(µ 1 ,M 11 ).Conditional Distributions All are normal. If X is as in the preceding sectionand X 2 is nondegenerate, then the conditional distribution of X 1 given X 2is normal withE(X 1 | X 2 )=µ 1 +M 12 M −122 (X 2 − µ 2 )var(X 1 | X 2 )=M 11 − M 12 M −122 M 21If X 2 is degenerate so M 22 is not invertible, then the conditional distributionof X 1 given X 2 is still normal and the same formulas work if M −122 is replacedby a generalized inverse.B.5.4The Bivariate Normal DistributionThe special case k = 2 of the preceeding section.The Density1f(x, y) = √2πσ X σ ×Y 1 − ρ2([1 (x − µX ) 2exp −2(1 − ρ 2 )σ 2 X− 2ρ(x − µ X)(y − µ Y )+ (y − µ Y ) 2 ])σ X σ Y σY2