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.

  • 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

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.


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