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Thread: Correlation between two normal variables ...

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    Correlation between two normal variables ...




    Suppose

    A ~ N (0,a),

    B ~ N (0,b-a),

    C = A + B

    what is the expression for the correlation between A and C?
    Last edited by BigBugBuzzz; 02-03-2012 at 12:43 PM.

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    Re: Correlation between two normal variables ...

    Are A and B independent? Nothing you posted has given information about the dependency between A and B.

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    Re: Correlation between two normal variables ...

    Sorry, yes, they are independent.

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    Re: Correlation between two normal variables ...

    Is this for homework? What have you tried so far?

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    Re: Correlation between two normal variables ...

    I attempted to start with cov(A,C) = E(AC) - E(A)E(C), but the problem is, I am sure the answer is not zero, but I am unsure how to show that. Intuitively when a = b, corr = 1 and when b --> inf corr --> 0.

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    Re: Correlation between two normal variables ...

    Quote Originally Posted by BigBugBuzzz View Post
    I attempted to start with cov(A,C) = E(AC) - E(A)E(C), but the problem is, I am sure the answer is not zero, but I am unsure how to show that. Intuitively when a = b, corr = 1 and when b --> inf corr --> 0.
    The correlation is not zero. And, I think you should define your variables as, say, X, Y, and Z so you don't get confused. Anyway, just use the definition of the correlation coefficient and the properties of expectation.

    Corr = Var[X] / ( Sqrt[ Var[X] + Var[Y - X] ] * Std[X])

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    Re: Correlation between two normal variables ...

    The following does not appear right...

    First, let A = X and C = Y as suggested.

    X ~ N (0,a)

    Y = X + N ~ (0,b-a) = N ~ (X,b-a)

    Now applying the formula given:


    Corr = Var[X] / ( Sqrt[ Var[X] + Var[Y - X] ] * Std[X]) -->

    Corr = a / ( sqrt(a + b) * sqrt(a) )

    I would have thought that when a = b, corr = 1, but that does not follow from my attempt...

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    Re: Correlation between two normal variables ...

    What did you put as Var[Y - X]? It doesn't look to me that you put the right quantity there.

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    Re: Correlation between two normal variables ...

    Var[Y - X] = b-a + a

    That is what I suspected was wrong. Do I not add the variances, even when subtracting two normal variables?

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    Re: Correlation between two normal variables ...

    In terms of the constants, I think you should end up with a correlation of:

    r=\frac{a}{\sqrt{2a^{2}-2ab+b^{2}}}

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    Re: Correlation between two normal variables ...

    I don't think that's quite right. I get r = a/sqrt(ab) as the final answer.

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    Re: Correlation between two normal variables ...

    X ~ N(0, a)
    Y ~ N(0, b - a)
    Z = X + Y

    I used Cov(X, X+Y) = Cov(X, X) + Cov(X, Y) = Var(X) + Cov(X, Y) = Var(X) by independence. Cor(X, Z) = Cov(X,Z)/sqrt(Var(X)Var(Z)).

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    Re: Correlation between two normal variables ...

    Dason, I just checked the equation I wrote empirically and it appears correct.

    In terms of the Var[z], did you use (b-a)^2 when you substituted?

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    Re: Correlation between two normal variables ...

    Var(Z) = b. Since X and Y are independent Var(X + Y) = Var(X) + Var(Y) = a + (b - a) = b.

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    Re: Correlation between two normal variables ...


    Quote Originally Posted by Dason View Post
    Var(Z) = b. Since X and Y are independent Var(X + Y) = Var(X) + Var(Y) = a + (b - a) = b.
    Ah, I think I see the problem now....you're treating a and b as variances and not standard deviations.

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