Cov(ax + b, cY + d) = - E(ax + b)E +, )-)) + adx + bcY + bd - ((аЕX) + b) + adE(X) + bcE(Y) + bd = + adE(X) + bcE - acE(X)E | - E(X)E = acCov(X, Y) p) Use part (a) along with the rules of variance and standard deviation to show that Corr(ax + b, cY + d) = Corr(X, Y) when a and c have the same sign. , CY + Cov Corr(ax + b, cY + d) = Oax + bº cY + d Y) lal|cloxoy Corr(X, Y) Ja||c| = Corr(X, Y) =) What happens if a and c have opposite signs? ас O when a and c differ in sign, ac is negative and = -1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). lallc| ас O When a and c differ in sign, ac is positive and - = 1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). la||c| ac O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). la||c| ас O When a and c differ in sign, ac is positive and = -1. Therefore, Corr(ax + b, CY + d) = -Corr(X, Y). Jallc| O When a and c differ in sign, ac is negative and ас = -1. Therefore, Corr(ax + b, cY + d) = Corr(X, ). Ja||c|
Cov(ax + b, cY + d) = - E(ax + b)E +, )-)) + adx + bcY + bd - ((аЕX) + b) + adE(X) + bcE(Y) + bd = + adE(X) + bcE - acE(X)E | - E(X)E = acCov(X, Y) p) Use part (a) along with the rules of variance and standard deviation to show that Corr(ax + b, cY + d) = Corr(X, Y) when a and c have the same sign. , CY + Cov Corr(ax + b, cY + d) = Oax + bº cY + d Y) lal|cloxoy Corr(X, Y) Ja||c| = Corr(X, Y) =) What happens if a and c have opposite signs? ас O when a and c differ in sign, ac is negative and = -1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). lallc| ас O When a and c differ in sign, ac is positive and - = 1. Therefore, Corr(ax + b, cY + d) = -Corr(X, Y). la||c| ac O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). la||c| ас O When a and c differ in sign, ac is positive and = -1. Therefore, Corr(ax + b, CY + d) = -Corr(X, Y). Jallc| O When a and c differ in sign, ac is negative and ас = -1. Therefore, Corr(ax + b, cY + d) = Corr(X, ). Ja||c|
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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