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Q2 Let (X₁, X₂) be jointly continuous with joint probability density function 2e (²1+2₂) 0 f(x1, x₂) = " 0 <₁ < ₂ <∞ otherwise. Q2(i.) Sketch(Shade) the support of (X₁, X₂). Q2(ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y2₂ = X₂ - X₁. Find the distribution of Y₂ using the distribution function method, i.e., find an expression for Fy₂ (y) = P(Y₂ ≤ y) = P(X₂ - X₁ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₂ - ₁ ≤y) and then find the probability density function of Y₂, i.e., fy₂ (y). Q2 (iv.) Using the bivariate transformation method, find the joint distribution of Y₁ = 2X₁ and Y₂ = X₂ - X₁. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(v.) Find the marginal density of Y₂ = X₂ X₁ and verify that it is the same density function obtained in part Q2(iii.).

Q2 Let (X₁, X₂) be jointly continuous with joint probability density function 2e (²1+2₂) 0 f(x1, x₂) = " 0 <₁ < ₂ <∞ otherwise. Q2(i.) Sketch(Shade) the support of (X₁, X₂). Q2(ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y2₂ = X₂ - X₁. Find the distribution of Y₂ using the distribution function method, i.e., find an expression for Fy₂ (y) = P(Y₂ ≤ y) = P(X₂ - X₁ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₂ - ₁ ≤y) and then find the probability density function of Y₂, i.e., fy₂ (y). Q2 (iv.) Using the bivariate transformation method, find the joint distribution of Y₁ = 2X₁ and Y₂ = X₂ - X₁. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(v.) Find the marginal density of Y₂ = X₂ X₁ and verify that it is the same density function obtained in part Q2(iii.).

Calculus For The Life Sciences
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 47E
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Q2 Let (X₁, X₂) be jointly continuous with joint probability density function 2e (²1+2₂) 0 f(x₁, x₂) = 2 0 < x1 < x₂ <∞ otherwise. Q2(i.) Sketch(Shade) the support of (X₁, X₂). Q2(ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y₂ = X₂ - X₁. Find the distribution of Y₂ using the distribution function method, i.e., find an expression for Fy₂ (y) = P(Y₂ ≤ y) = P(X₂ − X₁ ≤ y) using the joint probability density function (Hint: sketch or shade the region 2 - ₁ ≤y) and then find the probability density function of Y₂, i.e. fy₂ (y). Q2 (iv.) Using the bivariate transformation method, find the joint distribution of Y₁ = 2X₁ and Y₂ = X₂ - X₁. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2 (v.) Find the marginal density of Y₂ = X₂ - X₁ and verify that it is the same density function obtained in part Q2(iii.). Q2 (vi) Find the marginal density of Y₁₂ = 2X₁. Q2 (vii.) Are Y₁ and Y₂ independent?
Transcribed Image Text:Q2 Let (X₁, X₂) be jointly continuous with joint probability density function 2e (²1+2₂) 0 f(x₁, x₂) = 2 0 < x1 < x₂ <∞ otherwise. Q2(i.) Sketch(Shade) the support of (X₁, X₂). Q2(ii.) Are X₁ and X₂ independent random variables? Justify your answer. Identify the random variables X₁ and X₂. Q2 (iii.) Let Y₂ = X₂ - X₁. Find the distribution of Y₂ using the distribution function method, i.e., find an expression for Fy₂ (y) = P(Y₂ ≤ y) = P(X₂ − X₁ ≤ y) using the joint probability density function (Hint: sketch or shade the region 2 - ₁ ≤y) and then find the probability density function of Y₂, i.e. fy₂ (y). Q2 (iv.) Using the bivariate transformation method, find the joint distribution of Y₁ = 2X₁ and Y₂ = X₂ - X₁. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2 (v.) Find the marginal density of Y₂ = X₂ - X₁ and verify that it is the same density function obtained in part Q2(iii.). Q2 (vi) Find the marginal density of Y₁₂ = 2X₁. Q2 (vii.) Are Y₁ and Y₂ independent?
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