Answered: Q 8.4. Let X₁, X2, Y₁ and Y₂ be… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 31E

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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.

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