Let X and Y be random variables, and a and b be constants.

a. Prove that Cov(aX, bY) = ab Cov(X, Y).

b. Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.

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