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Homework Help: Variance and Covariance

  1. Mar 13, 2010 #1
    Very sorry that I've double posted but I realised i placed the original post in Precalculus.



    1. The problem statement, all variables and given/known data
    Question

    Let X and Y be independent random variables with variances 9 and 7 respectively and let
    Z = X - Y

    a) What is the value of Cov(X,Z)
    b) What is the value of the correlation coefficient of X and Z?

    I've been stuck on this one question for 2-3 hours; its ridiculous, I know. Here's my terrible try.


    3. The attempt at a solution
    a)

    Var(X) = 9
    Var(Y) = 7

    Var(X-Y) = Var(X) + Var(Y) = Var(Z)
    Therefore, Var(Z) = 7 + 9 =16
    Cov(X,Z) = E[XZ] - E[X]E[Z]


    and b) [tex]\rho[/tex]XZ = [tex]\frac{Cov(X,Z)}{\sqrt{Var(X)*Var(Z)}}[/tex]



    = [tex]\frac{Cov(X,Z)}{\sqrt{9}*\sqrt{16}}[/tex]
    = [tex]\frac{Cov(X,Z)}{12}[/tex]

    Since last topic, i've realised that Cov(X,Y) = 0 due to independency. But I don't know how to use it.
     
  2. jcsd
  3. Mar 13, 2010 #2

    Dick

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    Re: Variance/Covariance

    Cov(X,Z)=Cov(X,X-Y). Cov(X,X-Y)=Cov(X,X)-Cov(X,Y), right?
     
  4. Mar 13, 2010 #3
    Re: Variance/Covariance

    I didn't know about that law; thankyou very much.
     
  5. Mar 13, 2010 #4

    Dick

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    Re: Variance/Covariance

    It's pretty obvious if you write out the definition of Cov. You should try and do that so you can see why it's true.
     
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