Homework SourseLet X X1 X2 X3 have a multivariate normal distribution with
Let X = (X_1, X_2, X_3) have a multivariate normal distribution with mean vector vector 0 and variance-covariance matrix summation = [1 0 0 0 2 1 0 1 2] Find P(X_1 > X_2 + X_3 + 2).![Let X = (X_1, X_2, X_3) have a multivariate normal distribution with mean vector vector 0 and variance-covariance matrix summation = [1 0 0 0 2 1 0 1 2] Find P](/WebImages/29/let-x-x1-x2-x3-have-a-multivariate-normal-distribution-with-1081412-1761567941-0.webp "Let X = (X_1, X_2, X_3) have a multivariate normal distribution with mean vector vector 0 and variance-covariance matrix summation = [1 0 0 0 2 1 0 1 2] Find P")
Solution
from the variance - covariance matrix it is easy to see that X1 is independent of X2 and X3
hence
P(X1>X2+X3+2)+P(X1<X2+X3+2)=1
that is 2*P(X1>X2+X3+2)=1
that is P(X1>X2+X3+2))=1/2=0.5
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