Solution Found!
Let X and Y be random variables, and a and b be
Chapter 2, Problem 27E(choose chapter or problem)
Let X and Y be random variables, and a and b be constants.
a. Prove that \(\operatorname{Cov}(a X, b Y)=a b \operatorname{Cov}(X, Y)\).
b. Prove that if \(a>0\) and \(b>0\), then \(\rho_{a X, b Y}=\rho_{X, Y}\). Conclude that the correlation coefficient is unaffected by changes in units.
Equation Transcription:
Text Transcription:
Cov(aX,bY)=ab Cov(X,Y)
a>0
b>0
rho_{aX,bY}=rho_{X,Y}
Questions & Answers
QUESTION:
Let X and Y be random variables, and a and b be constants.
a. Prove that \(\operatorname{Cov}(a X, b Y)=a b \operatorname{Cov}(X, Y)\).
b. Prove that if \(a>0\) and \(b>0\), then \(\rho_{a X, b Y}=\rho_{X, Y}\). Conclude that the correlation coefficient is unaffected by changes in units.
Equation Transcription:
Text Transcription:
Cov(aX,bY)=ab Cov(X,Y)
a>0
b>0
rho_{aX,bY}=rho_{X,Y}
ANSWER:
Solution :
Step 1 of 2:
Let X and Y be the random variable.
Our goal is :
a). We need to prove that Cov(aX,bY)=abCov(X,Y).
b). We need to prove that if a>0 and b>0,then .
a).
Now we need to prove that Cov(aX,bY)=abCov(X,Y)
Here a and b are the constants.
Here
and
Then,
Here
Therefore