Consider a vector X X1 X2 X3 with normal distribution LX N

Consider a vector X = (X_1, X_2, X_3) with normal distribution L(X) = N(mux, Lambda ^2_x) with the following vector of means and Variance-Covariance Matrix; Find the distribution of the variables = X_1 - 2X_2 + 3X_3, The distribution of the vector (X_2 - X_3, X_1 - 2X_2, X_3), the conditional distribution of L ((X_1, X-2)|X_3 = 1), and the conditional distribution of L (X_1 - 2X-2| (X_2 - X_3 = 0, X_1)).

Solution

S = X1 - 2x2 + 3x3

As this variable is a linear combination of 3 normal independent variables

S is also normal with Mean = E(X1) -2E(x2) +3E(x3)=0+0-3 =-3

Var S= Var(x1)+ 4Var(X2) + 9 Var(X3)= 14 var (X)

= 14Var (x) = 14^_x²

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The distribution of (X2-X3,