Q1 Let (X₁, X₂) be jointly continuous with joint probability density function " f(x1, x₂) = x1 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., fx₂|x, (x₂|x1). Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X₂|X1 = x₁].
Q1 Let (X₁, X₂) be jointly continuous with joint probability density function " f(x1, x₂) = x1 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., fx₂|x, (x₂|x1). Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X₂|X1 = x₁].
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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![Q1 Let (X₁, X₂) be jointly continuous with joint probability density function
"
f(x1, x₂) = x1
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., ƒx₂|X₁ (x2|x1).
Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X2|X₁ = x1].
Q1(viii.) What is E[X₂|X₁]? 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 EX₁ - 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%2F635f13e0-d333-469d-8ee9-1fd6fc8e1a10%2Fkvc4ntl_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 Let (X₁, X₂) be jointly continuous with joint probability density function
"
f(x1, x₂) = x1
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., ƒx₂|X₁ (x2|x1).
Q1(vii.) Find the conditional expectation of X₂ given X₁ = x₁, i.e. E[X2|X₁ = x1].
Q1(viii.) What is E[X₂|X₁]? 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 EX₁ - X₂].
Q1(xi.) Using the joint density directly, find E[X₁ X₂], and the Cov(X₁, X₂).
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Step 1: Write the given information.
VIEWStep 2: Sketch the support of (X₁, X₂).
VIEWStep 3: Determine the marginal density function of x1 and x2.
VIEWStep 4: Determine the value of E(X1) and E(X2).
VIEWStep 5: Determine the conditional density of X2 given X1=x1.
VIEWStep 6: Determine the conditional expectation of X2 given X1=x1.
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