Q1 Let (X₁, X2) be jointly continuous with joint probability density function f(x1, x₂) = x1 0 " 0 < x2 < X1 < 1, otherwise. Q1(i.) Sketch(Shade) the support of (X₁, X₂). Q1(ii.) Find the marginal density function of X₁. Q1(iii.) Find the marginal density function of X2. Q1(iv.) Find E[X₁]. Q1(v.) Find E[X₂]. Q1(vi.) Find the conditional density of X₂ given X₁ = x₁, i.e., fx₂x₁ (₂₁). Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X₂|X1₁ = x1]. Q1(viii.) What is E[X₂X₁]? Using the property of conditional expectation, verify that E[X₂] is the same as the one obtained in part Q1(v.). Q1(ix.) Using the joint density directly, find E[X₂] and verify that it is the same as obtained Q1(v.) and Q1(viii.). Q1(x.) Using the joint density directly, find E[X₁ - X₂]. Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).
Q1 Let (X₁, X2) be jointly continuous with joint probability density function f(x1, x₂) = x1 0 " 0 < x2 < X1 < 1, otherwise. Q1(i.) Sketch(Shade) the support of (X₁, X₂). Q1(ii.) Find the marginal density function of X₁. Q1(iii.) Find the marginal density function of X2. Q1(iv.) Find E[X₁]. Q1(v.) Find E[X₂]. Q1(vi.) Find the conditional density of X₂ given X₁ = x₁, i.e., fx₂x₁ (₂₁). Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X₂|X1₁ = x1]. Q1(viii.) What is E[X₂X₁]? Using the property of conditional expectation, verify that E[X₂] is the same as the one obtained in part Q1(v.). Q1(ix.) Using the joint density directly, find E[X₂] and verify that it is the same as obtained Q1(v.) and Q1(viii.). Q1(x.) Using the joint density directly, find E[X₁ - X₂]. Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).
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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Solve parts 4,5,6
Thank you
![Q1 Let (X₁, X₂) be jointly continuous with joint probability density function
0 < x2 < x1 < 1,
otherwise.
f(x1, x₂)
=
x1
0
Q1(i.) Sketch(Shade) the support of (X₁, X₂).
Q1(ii.) Find the marginal density function of X₁.
Q1(iii.) Find the marginal density function of X₂.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = x₁, i.e., fx₂x₁ (x₂|x₁).
Q1(vii.) Find the conditional expectation of X₂ given X₁
x₁, i.e. E[X₂|X₁ = x₁].
=
Q1(viii.) What is E[X₂X₁]? Using the property of conditional expectation, verify that E[X₂] is the same as the
one obtained in part Q1(v.).
Q1(ix.) Using the joint density directly, find E[X₂] and verify that it is the same as obtained Q1(v.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ — X₂].
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%2Fffa92b76-a65d-440a-a5fe-72e3ee5c951e%2Fvk4tf0q_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 Let (X₁, X₂) be jointly continuous with joint probability density function
0 < x2 < x1 < 1,
otherwise.
f(x1, x₂)
=
x1
0
Q1(i.) Sketch(Shade) the support of (X₁, X₂).
Q1(ii.) Find the marginal density function of X₁.
Q1(iii.) Find the marginal density function of X₂.
Q1(iv.) Find E[X₁].
Q1(v.) Find E[X₂].
Q1(vi.) Find the conditional density of X₂ given X₁ = x₁, i.e., fx₂x₁ (x₂|x₁).
Q1(vii.) Find the conditional expectation of X₂ given X₁
x₁, i.e. E[X₂|X₁ = x₁].
=
Q1(viii.) What is E[X₂X₁]? Using the property of conditional expectation, verify that E[X₂] is the same as the
one obtained in part Q1(v.).
Q1(ix.) Using the joint density directly, find E[X₂] and verify that it is the same as obtained Q1(v.) and Q1(viii.).
Q1(x.) Using the joint density directly, find E[X₁ — X₂].
Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).
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