22 Let (X₁, X₂) be jointly continuous with joint probability density function е f(x₁, x2) = { 0-(²1+22), 21 > 0, 22 > 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. 22(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 nen find the probability density function of Y₁, i.e., fy, (y). 12(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function f Y₁, find E[Y₁]. 22(v.) Let Y2 = X1 — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y2, and using the moment generatin unction of Y₂, find E[Y₂]. (1-t) 2(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 Let (X₁, X₂) be jointly continuous with joint probability density function е f(x₁, x2) = { 0-(²1+22), 21 > 0, 22 > 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. 22(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 nen find the probability density function of Y₁, i.e., fy, (y). 12(iv.) Let Mx, (t) = Mx₂ (t) = (1¹), for t < 1. Find the moment generating function of Y₁, and using the moment generating function f Y₁, find E[Y₁]. 22(v.) Let Y2 = X1 — X2, and Mx, (t) = Mx₂ (t) = (1 t). Find the moment generating function of Y2, and using the moment generatin unction of Y₂, find E[Y₂]. (1-t) 2(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).
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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