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,
![](https://test.transtutors.com/qimg/ee8de425-e6cd-457c-9b66-56b2d93f7bcd.png)
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
![](https://test.transtutors.com/qimg/6926561f-26be-46d4-bc13-2ce64a4d9a42.png)
![](https://test.transtutors.com/qimg/d39e65ef-eca1-4a28-b175-2fa8a7dad798.png)