Question

5.30 Suppose two random variables Xand Yare both zero mean and unit variance. Furthermore, assume they have a correlation coefficient of p. Two new random variables are formed according to W = aX + bY, Z = cX + dY. Determine under what conditions on the constants a,b. c, and d the random variables W and Z are uncorrelated.

Answer 1

Here

Here E(X) = E(Y) = 0 and Var (X) = Var (Y) = 1 and corr (X, Y) = rho = > cov (X, Y) = rho Cov (W, Z) = Cov (aX + bY, cX + dY) Cov (W, Z) = ac Var (X) + ad Cov (X, Y) + bc Cov (X, Y) + bd Var (Y) Cov (W, Z) = ac * 1 + ad * rho + bc * rho + bd * 1 Cov (W, Z) = ac +bd + rho (ad + bc) Now, Cov (W, Z) = 0 ac + bd + rho (ad + bc) = 0 rho(ad + bc) = -(ac + bd) rho = -(ac + bd)/(ad + bc) If a, b, c and d are such that rho = -(ac + bd)/(ad + bc), then W and Z will be uncorrelated. Moreover, ac = -bd and ad = -bc, then also W and Z will be uncorrelated.

and

and $corr(X,Y) = \rho \Rightarrow cov(X,Y) = \rho$

$\Rightarrow Cov(W,Z) = acVar(X)+adCov(X,Y)+bcCov(X,Y)+bdVar(Y)$

$\Rightarrow Cov(W,Z) = ac*1+ad*\rho +bc*\rho +bd*1$

$\Rightarrow Cov(W,Z) = ac+bd+\rho (ad+bc)$

Now,

$\Rightarrow ac+bd+\rho (ad+bc)= 0$

$\Rightarrow \rho (ad+bc)= -(ac+bd)$

$\Rightarrow \rho= \frac{-(ac+bd)}{(ad+bc)}$

If a,b,c and d are such that $\rho= \frac{-(ac+bd)}{(ad+bc)}$ ,then W and Z will be uncorrelated.

Moreover, and ,then also W and Z will be uncorrelated.