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).
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).
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 52CR
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solve parts 7, 8, 9 please
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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 X₁ 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%2F558d7a6b-b2bc-4bb9-b0fb-7d7e73320f95%2F54q308h_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 X₁ 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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