Random vector X = (X1, X2, X3) T E R has multivariate normal distribution with parameters 2 4 -1 0 E(X) = (-1,1,2), Var(

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Random Vector X X1 X2 X3 T E R Has Multivariate Normal Distribution With Parameters 2 4 1 0 E X 1 1 2 Var 1 (92.18 KiB) Viewed 71 times

Random vector X = (X1, X2, X3) T E R has multivariate normal distribution with parameters 2 4 -1 0 E(X) = (-1,1,2), Var(X) = -1 1 0 0 0 2 (a) (2 points) Find P(2X2 > X1 +3). (b) (1.5 points)Find matrix A € R2x2 and vector u ERP such that for Y def A(X1, X3)T +p it holds YTY ~X(2), where x2 (2) denotes the chi-squared distribution with 2 degrees of freedom. (Hint: (a) cf. exercise 3.5.14 in the textbook: find vector a s.t. af x 2X2 – X1 and use the properties of the normal distribution. Similarly for part (b): find a linear transformation of (X1, X3)T that leads to the required distribution.)