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Math Help - A doubt on stastical independence , orthogonality and uncorrelatedness ?

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    A doubt on stastical independence , orthogonality and uncorrelatedness ?

    Hi friends,
    I wanted to make my concepts on statistical independence, uncorrelatedness and orthogonality clear. Suppose I have 2 random variables x and y. I have 2 pictures on the above concepts, which is more general picture? If you finds any mistake in the picture , please point it out.
    A doubt on  stastical independence , orthogonality and uncorrelatedness ?-ind_uncorr_orth1.jpg
    Picture (a)


    A doubt on  stastical independence , orthogonality and uncorrelatedness ?-ind_uncorr_orth2.jpg
    Picture (b)

    What is statistical independence and linear independent independence means? Are they same?

    Do pdf(x,y)=pdf(x)*pdf(y) always imply E(XY)=E(X)E(Y). Can any one please explain that?

    -Devanand T
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    Hey dexterdev.

    The second diagram is a better representation.

    Independent will always give you E[XY] = E[X]E[Y] and there can be overlap in orthogonality with independence. The orthogonal ones should always have the property of being un-correlated though (the covariance matrix should be diagonal for this case).
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    Hello Chiro,
    Are statistical independence and linear independent independence same ?
    How do pdf(x,y)=pdf(x)*pdf(y) always imply E(XY)=E(X)E(Y) ?

    -Thanks
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    Yes they are the same.

    For independence you have P(X = x, Y = y) = P(X = x)*P(Y=y).

    I thought about doing a derivation but the easiest way to prove this is to use what is known as Fubini's Theorem:

    Fubini's theorem - Wikipedia, the free encyclopedia

    Basically if you look at the corollary, just replace g(x) with x*P(X=x) and h(y) with y*P(Y=y) and thats the proof done.
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    thanks for the reply
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    Quote Originally Posted by chiro View Post
    Hey dexterdev.

    The second diagram is a better representation.

    Independent will always give you E[XY] = E[X]E[Y] and there can be overlap in orthogonality with independence. The orthogonal ones should always have the property of being un-correlated though (the covariance matrix should be diagonal for this case).

    Then what does this pdf says : (it has a different picture with linear independence etc)

    http://www.psych.umn.edu/faculty/wal...gs/rodgers.pdf
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    Re: A doubt on stastical independence , orthogonality and uncorrelatedness ?

    The above refers to observations: I was referring above to probabilistic independence (in terms of the independence of random variables through probabilistic properties) which is a little different.
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