Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂ and the marginal density of X2. and fx₁|x₂(x1x₂) = C1x1 2 0< x₁ <£₂, 0 < x₂ < 1 otherwise. fx.(2₂) = { 0₂² 0
Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂ and the marginal density of X2. and fx₁|x₂(x1x₂) = C1x1 2 0< x₁ <£₂, 0 < x₂ < 1 otherwise. fx.(2₂) = { 0₂² 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 29RE
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Could you solve parts 7, 8, and 9 please?
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![Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂
and the marginal density of X2.
and
fx₁|x₂(x1x₂)
=
C1x1
2
0 < x₁ <£₂, 0 < x₂ <1
otherwise.
fx.(2₂) = { 0₂² 0<x₂ <1,
otherwise.
where C₁ and C₂ are constants.
Q1(i.) Sketch(Shade) the support of (X₁, X₂) and find the constants ₁ and ₂
Q1(ii.) Find the joint probability density function of (X₁, X₂).
Q1(iii.) Find the marginal density function of X₁.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = ₁, i.e., fx₂x₁ (₂|x₁).
Q1(vii.) Find the conditional expectation of X₁ given X₂ = x2, i.e. E[X₁|X₂ = x₂].
Q1(viii.) What is E[X₁|X₂]? Using the property of conditional expectation, verify that E[X₁] is the same as the one obtained in part
Q1 (iv.).
Q1(ix.) Using the joint density directly, find EX₁] and verify that it is the same as obtained Q1(iv.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ X2₂].
Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F7943bbde-329b-413c-81b4-55d85bd3dbb4%2Fhze7k36_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 Let (X₁, X₂) be jointly continuous random vector with the following information: you are given the conditional density of X₁ given X₂
and the marginal density of X2.
and
fx₁|x₂(x1x₂)
=
C1x1
2
0 < x₁ <£₂, 0 < x₂ <1
otherwise.
fx.(2₂) = { 0₂² 0<x₂ <1,
otherwise.
where C₁ and C₂ are constants.
Q1(i.) Sketch(Shade) the support of (X₁, X₂) and find the constants ₁ and ₂
Q1(ii.) Find the joint probability density function of (X₁, X₂).
Q1(iii.) Find the marginal density function of X₁.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = ₁, i.e., fx₂x₁ (₂|x₁).
Q1(vii.) Find the conditional expectation of X₁ given X₂ = x2, i.e. E[X₁|X₂ = x₂].
Q1(viii.) What is E[X₁|X₂]? Using the property of conditional expectation, verify that E[X₁] is the same as the one obtained in part
Q1 (iv.).
Q1(ix.) Using the joint density directly, find EX₁] and verify that it is the same as obtained Q1(iv.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ X2₂].
Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).
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