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(a) Let U be a 4 x 4 matrix whose first two rows are (1/√/2 0 (1/√2 1/√2 0 0 1/√2 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!]

(a) Let U be a 4 x 4 matrix whose first two rows are (1/√/2 0 (1/√2 1/√2 0 0 1/√2 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!]

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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8.4 （a）
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 1 5- 2 The aim of this question is to describe the joint distribution of X - Y and S² Σ(Xi - X)² + (Y; - Y)² +2₁²-1²) i=1 i=1 (a) Let U be a 4 × 4 matrix whose first two rows are (1/√2 1/√2 1/√√2 0 0 1/√2 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!]
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 1 5- 2 The aim of this question is to describe the joint distribution of X - Y and S² Σ(Xi - X)² + (Y; - Y)² +2₁²-1²) i=1 i=1 (a) Let U be a 4 × 4 matrix whose first two rows are (1/√2 1/√2 1/√√2 0 0 1/√2 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!]
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