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Math Help - a question abt covariance

  1. #1
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    a question abt covariance

    Z~N(0,1), Let X1=Z, X2=Z^2, find cov(X1,X2).

    Cov(X1,X2)= E(X1*X2)-E(X1)E(X2)= E(Z^3)-E(Z)E(Z^2)= E(Z^3)
    bcoz E(Z)=0, does it mean E(Z^n)=0 ??? anyone can explain ?? thanx
    Last edited by solskjaer; November 22nd 2008 at 06:31 PM.
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  2. #2
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    Quote Originally Posted by solskjaer View Post
    Z~N(0,1), Let X1=Z, X2=Z^2, find cov(X1,X2).

    Cov(X1,X2)= E(X1*X2)-E(X1)E(X2)= E(Z^3)-E(Z)E(Z^2)= E(Z^3)
    bcoz E(Z)=0, does it mean E(Z^n)=0 ??? anyone can explain ?? thanx
    Let f(z) be the pdf of Z.

    Then E(Z^{2n+1}) = \lim_{a \rightarrow + \infty} \int_{-a}^{a} z^{2n+1} f(z) \, dz = 0 because z^{2n+1} f(z) is an odd function.
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