Let X N2(, ) where and define Y R2 by Y1 = X2 X1

Let X ∼ N2(ξ, Σ) where

and define Y ∈ R2 by Y1 = X2 − X1 and Y2 = X1 − 1.

a) Arguethat X has adensitywithrespecttoLebesguemeasureon R2.

b) Findtheconcentrationmatrix K of X.

c) Writeanexpressionforthedensity,andsketchthelevelcurves.

d) Whatisthedistributionof Y ?

e) Are Y1 and Y2 independent?

f) Giveanexpressionfortheprojectiononto L = {x ∈ R2 : x1 = 2x2} with respect totheconcentrationinnerproduct ⟨·, ·⟩K.

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