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Math Help - Correlation between two chi-squared random variables

  1. #1
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    Correlation between two chi-squared random variables

    Hi Folks!

    Let X1 have a chi-squared distribution with 1 degree-of-freedom.
    Let Xn have a chi-squared distribution with n degrees-of-freedom.

    Next, consider

    Z = X1+Xn

    Then, assuming that X1 and Xn are independent, Z has a chi-squared distribution with n+1 degrees-of-freedom.

    Are you able to provide any comments on how to obtain

    correlation(X1,Z)?

    Cheers!

    Der
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  2. #2
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    Re: Correlation between two chi-squared random variables

    Hey DerWundermann.

    You need to find the joint distribution of X1 and Z and then from that calculate Cov(X,Y) = E[XY] - E[X]E[Y]

    Remember that P(A|B) = P(A and B)/P(B) and that P(A) = Integral/Sum_out P(A|B=b)*P(B=b) for all appropriate b.
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  3. #3
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    Re: Correlation between two chi-squared random variables

    Many thanks for your comments, chiro! Very useful!
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  4. #4
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    Re: Correlation between two chi-squared random variables

    am i being a bit dim here or can you just note that the conditional correlation Cor(Z,X1|Xn) is 1 for all possible values of Xn?


    (i suspect im being a bit dim...so the question becomes, why not?)
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  5. #5
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    Re: Correlation between two chi-squared random variables

    My comments in post #4 are wrong, but i haven't figured out why yet.


    an alternative method which i think is valid and does not require calculating the joint distribution is:

    Corr(X_1,X_1 + X_n) = \frac{Cov(X1,X_1 + X_n)}{\sqrt{Var(X_1)} \sqrt{Var(X_1 + X_n))}}

    =\frac{Cov(X_1,X_1) + Cov(X_1,X_n)}{\sqrt{Var(X_1)}\sqrt{(Var(X_1) + Var(X_n))}}

    =\frac{Var(X_1) + 0}{\sqrt{Var(X_1)}\sqrt{(Var(X_1) + Var(X_n))}}

    =\sqrt{\frac{Var(X_1)}{Var(X_1) + Var(X_n)}

    =\sqrt{\frac{2}{2 + 2n}

    =\sqrt{\frac{1}{1 + n}
    Last edited by SpringFan25; July 1st 2013 at 04:21 PM.
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  6. #6
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    Re: Correlation between two chi-squared random variables

    Very clever!
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