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] = ₁, that E[Y₁] = E[Y₂] = 2 and that var(X₁): var (X₂) = var (Y₁) = var (Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = (Y₁ + Y₂). Define a random variable S² via 2 2 Σ(x − x)2 + Σ - 1)2 - Σ(X-8²) i=1 i=1 The aim of this question is to describe the joint distribution of X - Y and S² 82 = (a) Let U be a 4 x 4 matrix whose first two rows are (1/V2 √2 1/√2 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. 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- tribution. Suppose that E[X₁] = E[X2] = ₁, that E[Y₁] = E[Y₂] = 2 and that var(X₁): var (X₂) = var (Y₁) = var (Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = (Y₁ + Y₂). Define a random variable S² via 2 2 Σ(x − x)2 + Σ - 1)2 - Σ(X-8²) i=1 i=1 The aim of this question is to describe the joint distribution of X - Y and S² 82 = (a) Let U be a 4 x 4 matrix whose first two rows are (1/V2 √2 1/√2 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. 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
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
Related questions
Question
8.4（c）
![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] = 2 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
1/12 (2² (x₁ - X)² + 2(x − 8)²)
ΣΥ; -
i=1
i=1
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/√2 1/√2 0
(1/√2
1/√2)
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)T, and define V to be the
random vector V = UW. Find the mean vector and variance-covariance matrix of V.
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%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2F4ae9a1ee-cf04-45db-aa32-9de90cf72ac7%2Fpmn9cip_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] = 2 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
1/12 (2² (x₁ - X)² + 2(x − 8)²)
ΣΥ; -
i=1
i=1
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/√2 1/√2 0
(1/√2
1/√2)
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)T, and define V to be the
random vector V = UW. Find the mean vector and variance-covariance matrix of V.
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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