Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis- = μ2 and that var(X₁) = = tribution. Suppose that E[X₁] = E[X₂] = µ₁, that E[Y₁] = E[Y₂] = var (X₂) = var (Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Y = (Y₁+ Y₂). Define a random variable S² via 2 Σ(x; − x)2 + ΣΥ; - Y)2 - + 2(x-8²) i=1 i=1 1 +²7 x)² 2 The aim of this question is to describe the joint distribution of X - Y and S² (a) Let U be a 4 x 4 matrix whose first two rows are (¹/V² 1/√2 0 0 0 0 1/√2 1/√2) Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁ and V₂. Use this to describe the joint distribution of X - Y and S².
Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis- = μ2 and that var(X₁) = = tribution. Suppose that E[X₁] = E[X₂] = µ₁, that E[Y₁] = E[Y₂] = var (X₂) = var (Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Y = (Y₁+ Y₂). Define a random variable S² via 2 Σ(x; − x)2 + ΣΥ; - Y)2 - + 2(x-8²) i=1 i=1 1 +²7 x)² 2 The aim of this question is to describe the joint distribution of X - Y and S² (a) Let U be a 4 x 4 matrix whose first two rows are (¹/V² 1/√2 0 0 0 0 1/√2 1/√2) Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁ and V₂. Use this to describe the joint distribution of X - Y and S².
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis-
tribution. Suppose that E[X₁] = E[X₂]
µ₁, that E[Y₁] = E[Y₂]
μ2 and that var(X₁)
var (X₂) = var (Y₁) = var(Y₂) = o² > 0.
Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via
S²
=
2
Σ(Xi − X)² + Σ(Y₂ − Ỹ)²
ΣΥ
i-n²)
i=1
1 (2₁x
i=1
=
(1/√2 1/√2
0
The aim of this question is to describe the joint distribution of X - Y and S²
(a) Let U be a 4 4 matrix whose first two rows are
0
0
1/√2 1/√₂)
-
Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use
plenty of zeros!]
(b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the
random vector V = UW. Find the mean vector and variance-covariance matrix of V.
(c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁
and V₂. Use this to describe the joint distribution of X - Y and S².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53d0abe2-4cb5-4db5-bef8-ebd7977db539%2Fbb91b306-e2d4-4ff5-8542-6b2a95a291d1%2Ft4rgthm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis-
tribution. Suppose that E[X₁] = E[X₂]
µ₁, that E[Y₁] = E[Y₂]
μ2 and that var(X₁)
var (X₂) = var (Y₁) = var(Y₂) = o² > 0.
Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via
S²
=
2
Σ(Xi − X)² + Σ(Y₂ − Ỹ)²
ΣΥ
i-n²)
i=1
1 (2₁x
i=1
=
(1/√2 1/√2
0
The aim of this question is to describe the joint distribution of X - Y and S²
(a) Let U be a 4 4 matrix whose first two rows are
0
0
1/√2 1/√₂)
-
Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use
plenty of zeros!]
(b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the
random vector V = UW. Find the mean vector and variance-covariance matrix of V.
(c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁
and V₂. Use this to describe the joint distribution of X - Y and S².
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 39 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman