7G.5. Let the two random vectors X and Y be jointly Gaussian with E(X) = mx; E(Y) = my. E((X - mx)(X - mx)T) = Px E((X - mx)(Y - my)T) = Pxy E((Y - my)(Y - my)T) = Ry Show that the conditional distribution of X given Y is (X|Y) ~ N(mx + Pxy(Ry)^-1(Y - my), Px - Pxy(Ry)^-1PxyT)
Added by Mireia J.
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Therefore, we can write: p(X,Y) = (1/2π)^(n/2) |Σ|^(-1/2) exp{-1/2 ( [X-mx] [Y-my] ) Σ^(-1) ( [X-mx]T [Y-my]T )} where n is the dimension of X and Y, Σ is the covariance matrix of X and Y, and mx and my are the means of X and Y, respectively. Show more…
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(a)Show that a random vector X has a multivariate normal distribution if, and only if, every linear combination, θ · X = θ1X1 + θ2X2 + · · · + θnXn, θ ∈ R^n , of the components has a univariate normal distribution. (b) Suppose that a random vector X = (X1, X2, . . . , Xn) has a multivariate normal distribution. Use the previous result to show the following: (i) Show that Y = (X1, X2) also has a multivariate normal distribution. (ii) Let ai ∈ R be given constants and form the random vector Z = (a1X1 + a2X2, a3X3 + a4X4). Show that Z has a multivariate normal distribution.
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