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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|

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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(a) Use the rules of expected value to show that Cov(ax + b, cY + d) = ac Cov(X, Y). Cov(ax + b, CY + d) = E - E(ax + b)E ])-) + adx + bcY + bd - (aE(X) + b) CE = = acE + adE(X) + bcE(Y) + bd + adE(X) + bcE = acE acE(X)E 1)) = — асCov(X, Y) (b) 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. co( b, cY + Corr(ax + b, cY + d) = Oax + bº cY + d Cov(X, Y) la||clox0y Corr(X, Y) lallc| = Corr(X, Y) (c) What happens if a and c have opposite signs? ac O When a and c differ in sign, ac is negative 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). Jal|c ас O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lal|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 negative and = -1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lallc|
Transcribed Image Text:(a) Use the rules of expected value to show that Cov(ax + b, cY + d) = ac Cov(X, Y). Cov(ax + b, CY + d) = E - E(ax + b)E ])-) + adx + bcY + bd - (aE(X) + b) CE = = acE + adE(X) + bcE(Y) + bd + adE(X) + bcE = acE acE(X)E 1)) = — асCov(X, Y) (b) 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. co( b, cY + Corr(ax + b, cY + d) = Oax + bº cY + d Cov(X, Y) la||clox0y Corr(X, Y) lallc| = Corr(X, Y) (c) What happens if a and c have opposite signs? ac O When a and c differ in sign, ac is negative 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). Jal|c ас O when a and c differ in sign, ac is positive and = 1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lal|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 negative and = -1. Therefore, Corr(ax + b, cY + d) = Corr(X, Y). lallc|
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