Consider the vector X in Problem 5.7.1 and define the average to be Y = (X1 + X2 + X3)/3. What is the probability that Y > 4?
Problem
In this problem, we extend the proof of Theorem 5.16 to the case when A is m × n with m X = 0.
(a) Prove there exists an (n - m) × n matrix 
of rank n-m with the property that A' = 0.
(b) Let 
= 
-1X and define the random vector Use Theorem 5.16 for the case m = n to argue that Y¯ is a Gaussian random vector.
(c) Find the covariance matrix 
of 
. Use the result of Problem 5.7.8 to show that Y and Yˆ are independent Gaussian random vectors
Theorem
Given an n-dimensional Gaussian random vector X with expected valueµ X and covariance CX, and an m × n matrix A with rank(A) = m,
Is an m-dimensional Gaussian random vector with expected value µY = AµX + b and covariance CY = ACXA'
Problem 5.7.8
An n-dimensional Gaussian vector W has a block diagonal covariance matrix where CX is m × m, CY is (n - m) × (n - m). Show that W can be written in terms of component vectors X and Y in the form such that X and Y are independent Gaussian random vectors
