22 Let (X₁, X₂) be jointly continuous with joint probability density function f(x₁, x2) = { 0- (² e-(#₁+₂), x1 > 0, x₂ > 0 otherwise. 22(i.) Sketch(Shade) the support of (X₁, X2). 22(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 ¹y₁ (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and hen find the probability density function of Y₁, i.e. fy, (y). Q2(v.) Let Y₂ = X₁ – X2₂, and Mx₁ (t) = Mx₂ (t) = unction of Y₂, find E[Y₂]. 22(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]. (1-t)' =(1-1). Find the moment generating function of Y2, and using the moment generating 22(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). 22(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2(iii.). 22(viii.) Find the marginal density of Y₂ = X₁ - X₂.
22 Let (X₁, X₂) be jointly continuous with joint probability density function f(x₁, x2) = { 0- (² e-(#₁+₂), x1 > 0, x₂ > 0 otherwise. 22(i.) Sketch(Shade) the support of (X₁, X2). 22(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 ¹y₁ (y) = P(Y₁ ≤ y) = P(X₁ + X₂ ≤ y) using the joint probability density function (Hint: sketch or shade the region ₁ + x₂ ≤ y) and hen find the probability density function of Y₁, i.e. fy, (y). Q2(v.) Let Y₂ = X₁ – X2₂, and Mx₁ (t) = Mx₂ (t) = unction of Y₂, find E[Y₂]. 22(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]. (1-t)' =(1-1). Find the moment generating function of Y2, and using the moment generating 22(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). 22(vii.) Find the marginal density of Y₁ = X₁ + X₂ and verify that it is the same density function obtained in part Q2(iii.). 22(viii.) Find the marginal density of Y₂ = 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.CR: Chapter 13 Review
Problem 30CR
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