Show that if $Y = a X + b ( a \neq 0 ) \text {, then } \operatorname { Corr } ( X , Y ) = + 1 \text { or } - 1$ Under what conditions will $\rho = + 1 ?$

Name: Show that if $Y = a X + b ( a \neq 0 ) \text {, then } \operatorname { Corr } ( X , Y ) = + 1 \text { or } - 1$ Under what conditions will $\rho = + 1 ?$
Brand: Quizplus
Rating: 4.9 (32 reviews)

Question

Quizplus · Accepted Answer

The Answer is

\operatorname { Cov } ( X , Y ) = \operatorname { Cov } ( X , a X + b ) = E [ X \cdot ( a X + b ) ] - E ( X ) \cdot E ( a X + b ) = a V \operatorname { ar } ( X )

therefore
Corr

( X , Y ) = \frac { \operatorname { aVar } ( X ) } { \sqrt { \operatorname { Var } ( X ) \cdot \operatorname { Var } ( Y ) } } = \frac { \operatorname { aVar } ( X ) } { \sqrt { \operatorname { Var } ( X ) \cdot a ^ { 2 } \operatorname { Var } ( X ) } } = 1 ext { if } a > 0 ext {, and } - 1 ext { if } a < 0

Show That If Y=aX+b(a≠0), then Corr⁡(X,Y)=+1 or −1Y = a X + b ( a \neq 0 ) \text {, then } \operatorname { Corr } ( X , Y ) = + 1 \text { or } - 1Y=aX+b(a=0), then Corr(X,Y)=+1 or −1

Show That If $Y = a X + b ( a \neq 0 ) \text {, then } \operatorname { Corr } ( X , Y ) = + 1 \text { or } - 1$