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.

MATLAB: An Introduction with Applications
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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).
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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