Q1. In this question we will prove Proposition 4.1.3 from the notes. Suppose that X = (X₁,..., Xp) is a random vector taking values in RP, and that E[[X²] <∞. (a) Prove that we may write Cov(X) = E[(X-EX)(X - EX)¹]. (b) Citing any results you use from the notes, prove that Tr(vv) = |v|² for any v ERP. (c) Using (a) and (b), show that Р E|X - EX² = Var(X;) = Tr(Cov(X)). i=1
Q1. In this question we will prove Proposition 4.1.3 from the notes. Suppose that X = (X₁,..., Xp) is a random vector taking values in RP, and that E[[X²] <∞. (a) Prove that we may write Cov(X) = E[(X-EX)(X - EX)¹]. (b) Citing any results you use from the notes, prove that Tr(vv) = |v|² for any v ERP. (c) Using (a) and (b), show that Р E|X - EX² = Var(X;) = Tr(Cov(X)). i=1
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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N2,
![Q1. In this question we will prove Proposition 4.1.3 from the notes. Suppose that X = (X1,..., Xp)T is
a random vector taking values in RP, and that E[|X|²] < ∞.
(a) Prove that we may write
Cov(X) = E[(X – EX)(X – EX)"].
-
(b) Citing any results you use from the notes, prove that
Tr(vv") = |v|
%3D
for any v E RP.
(c) Using (a) and (b), show that
E[X – EX|2 = E Var(X;) = Tr(Cov(X).
i=1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d63c4c0-1d60-4fb3-84fd-f3eaa9e9798b%2F436beb0e-ca35-435d-8ce2-22949a67b479%2F2of05xn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q1. In this question we will prove Proposition 4.1.3 from the notes. Suppose that X = (X1,..., Xp)T is
a random vector taking values in RP, and that E[|X|²] < ∞.
(a) Prove that we may write
Cov(X) = E[(X – EX)(X – EX)"].
-
(b) Citing any results you use from the notes, prove that
Tr(vv") = |v|
%3D
for any v E RP.
(c) Using (a) and (b), show that
E[X – EX|2 = E Var(X;) = Tr(Cov(X).
i=1
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