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)
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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Question
![(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|](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2Fd2eabe13-ee40-47e5-8b7d-8c8a07b0728b%2F9qkngx5_processed.png&w=3840&q=75)
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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