Q1 Let (X₁, X₂) be jointly continuous with joint probability density function -={₁ x1 0 f(x1, x₂) 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. 0 < x2 < X1 <1, otherwise.
Q1 Let (X₁, X₂) be jointly continuous with joint probability density function -={₁ x1 0 f(x1, x₂) 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. 0 < x2 < X1 <1, otherwise.
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 29CR
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![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%2F8c2eb526-bfd8-4bdd-95fb-b555f15feb04%2Fr0ijzb_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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