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Why is correlation coefficient -1 to +1

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Why is the correlation coefficient between -1 and +1?


    2. Relevant equations
    we know correlation coefficient
    [tex] \rho = \frac{E[xy]-E[x]E[y]}{\sqrt{\sigma_{x} \sigma_{y}}} [/tex]

    OR

    [tex] r = \frac{\sum ^n _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^n _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^n _{i=1}(Y_i - \bar{Y})^2}} [/tex]

    3. The attempt at a solution
    Is there a way to prove this analytically? Perhaps we can use the second formula and prove by induction the bottom is greater than the top? Or perhaps equal??

    I tried using the expected value formula for the first version with rho - i couldnt really use that properly. Canyou please suggest an approach? Can this even be done analytically? Or would it just have to be explained?

    Thanks for your help!!
     
  2. jcsd
  3. Mar 24, 2014 #2

    Ray Vickson

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    Your formula for ##\rho## is incorrect; it should be
    [tex] \rho \equiv \frac{E(X-EX)(Y-EY)}{\sigma_X \sigma_Y} = \frac{E(XY) - EX EY}{\sigma_X \sigma_Y} [/tex] You should not have ##\sqrt{ \;\; }## in the denominator.
     
  4. Mar 24, 2014 #3

    Ray Vickson

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    Since
    [tex] 0 \leq \text{Var}(aX+bY) = a^2 \sigma_X^2 + 2 a b\, \text{Cov}(X,Y) + b^2 \sigma_Y^2,[/tex]
    for all ##a,b##, and since ##\text{Cov}(X,Y) = \sigma_X \sigma_Y \, \rho##, the matrix
    [tex] M = \pmatrix{\sigma_X^2 & \sigma_X \sigma_Y \, \rho\\
    \sigma_X \sigma_Y \, \rho & \sigma_Y^2} [/tex]
    must be positive semidefinite. Apply the standard tests for semidefinitness.
     
  5. Mar 24, 2014 #4

    jbunniii

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    This is a direct consequence of the Cauchy-Schwarz inequality, a very important result which shows up in many forms throughout mathematics. There is a proof on the Wiki page in the context of an abstract inner-product space. It's a good exercise to go through the proof as written, and then restate the result by translating the concepts of inner product and norm into the language of probability.
     
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