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).

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

Section13.1: Continuous Probability Models

Problem 8E

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