0
$\begingroup$

So I have two random variables, X and Y. And their scale location transformations are :

U = a*X + b

W = c*Y + d

I'm given that a,b,c and d are constants. And their mean folow:

E(U) = a*E(X)+b

E(W) + c*E(Y)+d

Now I'm asked to prove that their covariance folows:

V(U,W) = acV(X,Y)

and that

V(U+W) = a^2*V(X) + c^2*V(Y) + 2acV(X,Y)

And I can't for the life of me figure out how.

$\endgroup$
4
  • 1
    $\begingroup$ Maybe start with ${\rm cov}(X,Y) = E(XY) - E(X)E(Y)$ and see where that gets you.... $\endgroup$ Jan 17 '17 at 17:43
  • $\begingroup$ I tried, but the only answer I found is that E(XY) = E(X)*E(Y) which results in 0. $\endgroup$ Jan 17 '17 at 17:55
  • $\begingroup$ Try again. Compute $E(UW) - E(U)E(W)$ after substituting in $U = aX + b$ and $W = cY + d$ and you'll get something in terms of $a,b,c,d$ and ${\rm cov}(X,Y)$. $\endgroup$ Jan 17 '17 at 17:57
  • $\begingroup$ I'm very sorry if I sound stupid, but could you write that out for me? Because this is pretty much where I get stuck I think. $\endgroup$ Jan 17 '17 at 18:00
0
$\begingroup$

Here's the first part:

enter image description here

For the second part (assuming you can't use the formula for the variance of a sum) do the exact same thing while remembering that ${\rm var}(Z) = {\rm cov}(Z,Z)$ for any random variable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.