Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 29J / a

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- Solution a

Solution

Assume

V_{YY}

is invertible. Put

R = X - μ_{X} - V_{X Y} V_{YY}^{- 1} (Y - μ_{Y}) .

This and

Y

are jointly multivariate normal, with

Cov (R, Y) = 0

and

Cov (R) = V_{XX} - V_{X Y} V_{YY}^{- 1} V_{Y X}

. Their joint Gaussian characteristic function factors when the cross covariance vanishes, so they are independent. Conditioning on

Y = y

therefore leaves the law of

R

unchanged, proving

X ∣ Y = y \sim N (μ_{X} + V_{X Y} V_{YY}^{- 1} (y - μ_{Y}), V_{XX} - V_{X Y} V_{YY}^{- 1} V_{Y X}) .

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