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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 44E
Related questions
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![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.
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