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Thread: Implications of Joint Independence

  1. #1
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    Implications of Joint Independence

    Hi, I have a question regarding the implications of joint independence. There are 3 random variables U,V and Z.

    (U,V) have a bivariate normal distribution are are jointly independent of Z. First does this imply that U is independent of Z and V is independent of Z separately?

    If not, what is the difference between saying U independent of Z and V independent of Z as opposed to (U,V) jointly independent of Z?

    Thank you!

  2. #2
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    Re: Implications of Joint Independence

    I think the independence of random vectors is defined by

    f_{U,V,Z}(u, v, z) = f_{U, V}(u, v)f_Z(z) ~~ \forall u, v, z \in \mathbb{R}

    So if you integrate with respect to u, we have

    \int_{-\infty}^{+\infty} f_{U,V,Z}(u, v, z) du = \int_{-\infty}^{+\infty} f_{U, V}(u, v)f_Z(z) du

    \Rightarrow f_{V,Z}(v,z) = f_V(v)f_Z(z) ~~ \forall v, z \in \mathbb{R}

    which implies V,Z are independent. The same holds for U, Z and thus your assertion is correct.

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