Stat 5101 Lecture Notes - School of Statistics

162 Stat 5101 (Geyer) Course Notes5-10. Specialize the formula (5.24) for the non-degenerate multivariate normaldensity to the two-dimensional case, obtaining1f(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 σY2Hint: To do this you need to know how to invert a 2 × 2 matrix and calculateits determinant. If( )a11 aA =12a 21 a 22thendet(A) =a 11 a 22 − a 12 a 21and( )a22 −a 12A −1 −a 21 a 11=det(A)(This is a special case of Cramer’s rule. It can also be verified by just doing thematrix multiplication. Verification of the formulas in the hint is not part of theproblem.)5-11. Specialize the conditional mean and variance in Theorem 5.15 to thetwo-dimensional case, obtainingE(X | Y )=µ X +ρ σ Xσ Y(Y −µ Y )var(X | Y )=σ 2 X(1 − ρ 2 )5-12 (Ellipsoids of Concentration). Suppose X is a non-degenerate normalrandom variable with density (5.24), which we rewrite ase −q(x)/2f(x) =(2π) n/2 det(M) 1/2A level set of the density, also called a highest density region is a set of the formS = { x ∈ R n : f(x) >c}for some constant c. Show that this can also be writtenS = { x ∈ R n : q(x)