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Linear Correlation

  1. Oct 15, 2007 #1
    I am going through some of the problems in a statistical physics book. I am stuck on the following question:

    If we have two random variables X and Y, related by Y=mX + b => y(i) = mx(i) + b, where b and m are deterministic, prove that corr(X,Y) = m/sqrt(m^2) = sgn(m), where sgn(m) is the sign function of m.

    I can see that this is perfect linear correlation, but I am not sure as to what is the obvious proof....I know that corr(X,Y) = (E(XY) - E(X)E(Y))/sd(X)*sd(Y)....

    Is the following derivation even logically valid?
    E(XY)-E(X)E(Y)
    = E(X(mX+b)) - E(X)E(mX+b)
    =E(mX^2 + bX)-E(X)E(mX+b)
    =mE(X^2)+bE(X)-mE(X)E(X)+b
    =m(E(X^2)-E(X)E(X))+bE(X)+b
    =m(Var(X))+bE(X) +b

    Thanks!
     
  2. jcsd
  3. Oct 15, 2007 #2

    mathman

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    Gold Member

    Your approach is valid. Your arithmetic is wrong.

    E(XY)-E(X)E(Y)=m(Var(X)). The b terms cancel when you do it right!

    The denominator will be |m|Var(X).
     
  4. Oct 16, 2007 #3
    Thank you!
     
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