Let X and Y be random variables defined on the same sample s

Let X and Y be random variables defined on the same sample space and having finite variances, and let a and b be real numbers. Show that Cov(aX + bY, aX - bY) = a^2 Var(X) - b^2 Var(Y). Conclude that, if X and Y have equal variances, then X + Y and X - Y are uncorrelated random variables.

Solution

from the definition of covariance we have

COV(X,Y)=E[XY]-E[X]*E[Y]   where E[X] denotes the expectation of X

a) so COV(aX+bY,aX-bY)=E[(aX+bY)*(aX-bY)]-E[aX+bY]*E[aX-bY]

                                =E[a²X²+abXY-abXY-b²Y²]-[aE[X]+bE[Y]]*[aE[X]-bE[Y]]

                                =E[a²X²]-E[b²Y²]-{a²E[X]²+abE[X]E[Y]-abE[X]E[Y]-b²E[Y]²}

                                =a²E[X²]-a²E[X]²-{b²E[Y²]-b²E[Y]²} [ a and b are real constants]

                                =a²(E[X²]-E[X]²)-b²(E[Y²]-E[Y]²)

                                =a²Var(X)-b²Var(Y) [proved]

[as Var(X)=E[X²]-E[X]²   and Var(Y)=E[Y²]-E[Y]²   are finite ]

b) X and Y have equal variances. so Var(X)=Var(Y)

X and Y are said to be uncorrelated if Cov(X,Y)=0

putting a=1 and b=1 in the previous identity then we have

Cov(X+Y,X-Y)=1²Var(X)-1²Var(Y)=Var(X)-Var(Y)=0    [as they are equal]

hence X and Y are uncorrelated random variables [proved]