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 Y = (Y₁+Y2). Define a random variable S² via 2 2 82 = 1 2 Σ(X₁-X)² + Σ(Yi-Y)² 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 0 (1/V/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!] (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. a): U=₁ OM OML OMLOMI- 。E。 ON TIN
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 Y = (Y₁+Y2). Define a random variable S² via 2 2 82 = 1 2 Σ(X₁-X)² + Σ(Yi-Y)² 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 0 (1/V/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!] (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. a): U=₁ OM OML OMLOMI- 。E。 ON TIN
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we have the orthogonal matrix of U from (a), please calculate the (b) thanks :)
![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 = (Y₁ + Y₂). Define a random variable S² via
S²
=
2
i=1
(Xi − X)² + Σ(Yi – Ý)²
+ 2M-1²)
-
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
0
(¹/√2
1/√2)
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!]
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.
(a): U =
OMILOMI
-I TIL D
31-01-0
جال م ا ه](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2Ffbd5409b-87ab-4b8b-b662-82e0e79338cf%2Fcbenkxw_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 = (Y₁ + Y₂). Define a random variable S² via
S²
=
2
i=1
(Xi − X)² + Σ(Yi – Ý)²
+ 2M-1²)
-
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
0
(¹/√2
1/√2)
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!]
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.
(a): U =
OMILOMI
-I TIL D
31-01-0
جال م ا ه
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