How to prove the following equations, assuming that YY and X1,X2,…XnX_1,X_2,…X_n are jointly normally distributed?

E(Y|X)=E(Y|X1,X2,…,Xn)=μY+Cov(Y,X)TVar(X)−1(X−EX), E(Y|{\bf X})=E( Y|X_1,X_2,…,X_n) =\mu_Y + Cov(Y,{\bf X})^T Var({\bf X})^{-1}({\bf X}-E{\bf X}),

Var(Y|X)=Var(Y|X1,X2,…,Xn)=σ2Y−Cov(Y,X)TVar(X)−1Cov(Y,X). Var(Y|{\bf X})=Var(Y|X_1,X_2,…,X_n) =\sigma_Y^2 -Cov(Y,{\bf X})^T Var({\bf X})^{-1}Cov(Y,{\bf X}).

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