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] = µ2 and that var(X₁) = var(X₂) = var(Y₁) = var(Y₂) = 0² > 0. Let X = (X₁ + X₂) and Ỹ = (Y₁ + Y₂). Define a random variable S² via 82 - (2 = i=1 The aim of this question is to describe the joint distribution of X - Y and S² 2 Σ(Xi − X)² + Σ(Y₁ – Ý)² + Σm-n²) i=1
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] = µ2 and that var(X₁) = var(X₂) = var(Y₁) = var(Y₂) = 0² > 0. Let X = (X₁ + X₂) and Ỹ = (Y₁ + Y₂). Define a random variable S² via 82 - (2 = i=1 The aim of this question is to describe the joint distribution of X - Y and S² 2 Σ(Xi − X)² + Σ(Y₁ – Ý)² + Σm-n²) i=1
MATLAB: An Introduction with Applications
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Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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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
8² - 1 (21X₁-RP+ (M-91²)
82
=
2
2
i=1
i=1
The aim of this question is to describe the joint distribution of X – Ỹ and S²
(a) Let U be a 4 x 4 matrix whose first two rows are
(1/1/2
√√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.
(c) Show that you can write S² as a function of V3 and V4 and that Ă – Ỹ 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%2F4a97bb2c-6d83-4604-9a70-f559a7370826%2F47ce8b5f-46fb-4427-8645-ee4cd50b211d%2Fm7yuyj_processed.png&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
8² - 1 (21X₁-RP+ (M-91²)
82
=
2
2
i=1
i=1
The aim of this question is to describe the joint distribution of X – Ỹ and S²
(a) Let U be a 4 x 4 matrix whose first two rows are
(1/1/2
√√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.
(c) Show that you can write S² as a function of V3 and V4 and that Ă – Ỹ is a function of V₁
and V₂. Use this to describe the joint distribution of X – Y and S².
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