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Math Help - Multivariate normal distribution and marginal distribution

  1. #1
    Junior Member
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    Multivariate normal distribution and marginal distribution

    Hi everyone,
    I have the following exercise:
    Given Y \sim \mathcal{N}_p(\mu,\Omega ) ,

    a) Consider the following decomposition Y=(Y_1,Y_2)^T, \mu=(\mu_1, \mu_2)^T, \Omega=( \Omega_{11}, \Omega_{12};\Omega_{21},\Omega_{22} ) ( omega is supposed to be a matrix).
    Show that conditional Y_1 |(Y_2=y_2) is \mathcal{N}_p ( \mu_1+\Omega_{12}\Omega_{22}^{-1}(y_2-\mu_2),\Omega_{11}-\Omega_{12}\Omega_{22}^{-1}\Omega_{21}), where p is the dimension of Y_1.

    This one, I have shown.

    b) Let a,b \in \mathbb{R}^n. Find the conditional X_1|X_2=x_2 where X_1=a^TY,X_2=b^TY. In which case this distribution doesn't depend on x_2?

    This one is causing me trouble. I stated by writing explicitly the product in f_{X}(a^TY|b^TY=x_2) but it gets me nowhere.

    Thank you in advance for taking time to answer my question.
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  2. #2
    MHF Contributor
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    Re: Multivariate normal distribution and marginal distribution

    Hey sunmalus.

    One suggestion I have is to use your results in a) and find the situation where the conditional distribution is free from any value of x2. (Think of when sigma_12 * sigma_22_inverse = 0)
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  3. #3
    Junior Member
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    Re: Multivariate normal distribution and marginal distribution

    Well, with some linear transformation ( (a^T, b^T)^T*Y=(X_1, X_2))I found the conditional distribution for b) but I have some atrocious matrix multiplication to do to find the exact form of my new matrix Omega in terms of a and b and the old Omega. I'm really wondering if there isn't another way. Plus my answer for last part is, as chiro said, when sigma_12 * sigma_22_inverse = 0. But this implies a lot of ugly sub cases... what am I missing, I don't think it should be as messy as what I've found.
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