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Math Help - bivariate normal distribution...

  1. #1
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    bivariate normal distribution...

    if (X,Y) have bivariate normal distribution N_2(0,S), S = (s_ij)
    s_11 = s_22 = 1 and s_12 = s_21 = p (correlation coefficient)

    is it correct to say that X,Y are both normal with N(0,1)?

    furthermore, is it right to say that since p = cov(X,Y) / (s_1 * s _2) <=>

    p = cov(X,Y) = E(XY) + 0 + 0 + 0 = E(x^2) = 1

    ???
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  2. #2
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    Quote Originally Posted by sezmin View Post
    if (X,Y) have bivariate normal distribution N_2(0,S), S = (s_ij)
    s_11 = s_22 = 1 and s_12 = s_21 = p (correlation coefficient)

    is it correct to say that X,Y are both normal with N(0,1)?
    Yes

    furthermore, is it right to say that since p = cov(X,Y) / (s_1 * s _2) <=>

    p = cov(X,Y) = E(XY) + 0 + 0 + 0 = E(x^2) = 1

    ???
    No ! Why would E[XY]=E[X^2] ??
    Let's come to think about that : if it was possible to compute the covariance, they wouldn't have put p. You just can't compute it with the information you're given.
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  3. #3
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    Ah, ok. Thanks!

    Then...if I had an arbitrary function g(t) defined on [0, infinity)

    E[ h(X^2) * (Y^2) ] =/= E[ h(X^2) * (X^2) ]

    correct?
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  4. #4
    Moo
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    Yes, it sure is different ! (unless you have very particular conditions)
    The correlations of X and X, and of X and Y are not always the same, in particular !
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  5. #5
    MHF Contributor matheagle's Avatar
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    The correlation of X and X is 1.
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