Question

Suppose (X, Y ) has bivariate normal distribution, E(X) = E(Y ) = 0,V ar(X) = σX2 , V ar(Y ) = σY2 and Correl(X, Y ) = ρ. Calculate the conditional expectation E(X2|Y ).

I. Suppose (X,Y) has bivariate normal distribution, E(X) = E(Y) 0, Var(X)-σ , Var(Y) σ and Correl (X,Y)-p. Calculate the conditional expectation ECKY expectation E(X2Y)

Answer 1

Since (X, Y ) have a bivariate normal distribution with , E(X) = E(Y ) = 0,V ar(X) = σ²_X , V ar(Y ) = σ²_Y and Correlation coefficient ρ, the conditional distribution of X given Y is univariate normal with conditional mean

E(X|Y)= Yρ σ_X/ σ_Y and conditional variance Var(X|Y)= σ²_X (1-ρ²).         …………………….(*)                

                                                                                   

Then E(X²|Y )=[ E(X²|Y )- {E(X|Y)}²]+{E(X|Y)}²

                      =Var(X|Y)+ {E(X|Y)}²

                              ⁼ σ²_X (1-ρ²)+{ Yρ σ_X/ σ_Y}²   (using (*))

=    σ²_X( 1-ρ²+ ρ²Y²/ σ²_Y)