Q1 Let (X₁, X₂) be jointly continuous with joint probability density function 1 { x1 f(x1, x₂) = 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 (x₂x₁). 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 par 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 1 { x1 f(x1, x₂) = 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 (x₂x₁). 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 par 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).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Could you please solve it by hand? And please add detailed explanations
And for the i) please draw it on the graph (do not just describe it with words)
Thank you
![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 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%2Fc743d384-ba77-41f2-82d5-4b94d49993e1%2Fwx654y_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 E[X₁ - 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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