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Property of correlation coefficient

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove the following:
    Let [tex]\rho[/tex] be the correlation coefficient.
    [tex] \rho(X, Y) = 1 \iff P(Y=aX+b) = 1[/tex]

    2. Relevant equations
    [tex] \rho(X, Y) = cov(X,Y)/\sigma_x\sigma_y [/tex]

    3. The attempt at a solution
    I have no idea how to prove this, I know that rho must lie between 1 and -1 (inclusive) and that values close to 1 indicate that high values of X must go with high values of Y. But I don't know how to formally prove the above problem.
  2. jcsd
  3. Mar 19, 2012 #2
    bump. I can't believe nobody hasn't proven or seen the proof of this.
  4. Mar 19, 2012 #3


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    Homework Helper

    As this is an if and only if, you must prove both ways, I suggest starting with
    [tex] P(Y=aX+b)=1 \ \ \implies \ \ \rho(X,Y) = 1[/tex]

    Now as the probability is one, you can start with
    [tex] Y=aX+b[/tex]

    Now assume you know [itex] \mu_X, \sigma_x [/itex] and calculate [itex] \rho(X,Y)[/itex]

    The other direction may be slightly trickier, but you should get some good insights from the first exercise
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