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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

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

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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 EX₁ X₂], and the Cov(X₁, X₂).
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 EX₁ X₂], and the Cov(X₁, X₂).
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