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Q2 Let (X₁, X2) be jointly continuous with joint probability density function f(x1, x2) = { 0 -(²₁+₂), X₁ > 0, x₂ > 0 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 Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + ₂ ≤y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. Q2(v.) Let Y2 = X₁ — X2, and Mx, (t) = Mx₂ (t) = (1-1). Find the moment generating function of Y2, and using the moment generating Function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ – X₂. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.). 02(viii.) Find the marginal density of Y₂ = X₁ X₂.

Q2 Let (X₁, X2) be jointly continuous with joint probability density function f(x1, x2) = { 0 -(²₁+₂), X₁ > 0, x₂ > 0 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 Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + ₂ ≤y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. Q2(v.) Let Y2 = X₁ — X2, and Mx, (t) = Mx₂ (t) = (1-1). Find the moment generating function of Y2, and using the moment generating Function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ – X₂. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.). 02(viii.) Find the marginal density of Y₂ = X₁ X₂.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 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 Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ − X₂. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.). Q2 (viii.) Find the marginal density of Y₂ = X₁ X₂.
Transcribed Image Text:Q2 Let (X₁, X₂) be jointly continuous with joint probability density function = { -(x₁+x2), f(x₁, x₂) = x1 > 0, x₂ > 0 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 Y₁ = X₁ + X₂. Find the distribution of Y₁ using the distribution function method, i.e., find an expression for Fy, (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and then find the probability density function of Y₁, i.e., fy, (y). Q2(iv.) Let Mx, (t) = Mx₂ (t) = (¹, for t < 1. Find the moment generating function of Y₁, and using the moment generating function of Y₁, find E[Y₁]. 1 - Q2(v.) Let Y₂ = X₁ – X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y₂, and using the moment generating function of Y₂, find E[Y₂]. Q2(vi.) Using the bivariate transformation method, find the joint distribution of Y₁ = X₁ + X₂ and Y₂ = X₁ − X₂. Sketch the support of (X₁, X₂) and (Y₁, Y₂) side by side and clearly state the support for (Y₁, Y₂). Q2(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2 (iii.). Q2 (viii.) Find the marginal density of Y₂ = X₁ X₂.
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