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Math Help - marginal distributions

  1. #1
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    marginal distributions

    Hey,

    I am working on this problem and I am stuck because of my lack of understanding of marginal and conditional distributions.

    We have two random vectors x and z which both have gaussian distributions defined as: N(Ux, Cx) and N(Uz,Cz) (where Ux and Uz are the means and Cx and Cx are covariance matrices). We define a third vector y such that y = x + z. The problem is to find the marginal distribution of y.

    I believe I can assume the vector y has a gaussian distribution with the following characteristics, N(Ux + Uz, Cx + Cz). Essentially I am saying the sum of two random vectors having gaussian distributions will create a new random vector following a gaussian distribution where the mean and covariance is the sum of the means and covariances of the individual vectors.

    Even if my distribution for y is correct, I dont understand what is meant by the "marginal distribution of y" which is what I am supposed to find an expression for.

    Any hints or help with understanding these concepts.

    Thanks
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  2. #2
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    Quote Originally Posted by eigenstein22 View Post
    Hey,

    I am working on this problem and I am stuck because of my lack of understanding of marginal and conditional distributions.

    We have two random vectors x and z which both have gaussian distributions defined as: N(Ux, Cx) and N(Uz,Cz) (where Ux and Uz are the means and Cx and Cx are covariance matrices). We define a third vector y such that y = x + z. The problem is to find the marginal distribution of y.

    I believe I can assume the vector y has a gaussian distribution with the following characteristics, N(Ux + Uz, Cx + Cz). Essentially I am saying the sum of two random vectors having gaussian distributions will create a new random vector following a gaussian distribution where the mean and covariance is the sum of the means and covariances of the individual vectors.

    Even if my distribution for y is correct, I dont understand what is meant by the "marginal distribution of y" which is what I am supposed to find an expression for.

    Any hints or help with understanding these concepts.

    Thanks
    When you say I believe, I hope you mean the following reasoning:
    A gaussian vector's distribution is purely determined by its mean vector and covariance matrix.

    y = x+z \implies Uy = \mathbb{E}(y) = \mathbb{E}(x+z) = Ux + Uz

    y = x+z \implies Cy = \mathbb{E}((y - Uy)(y - Uy)^T) = \text{Cov }(x,x) + \text{Cov }(x,z)  + \text{Cov }(z,x) + \text{Cov }(z,z) = Cx + 0 + 0 + Cz

    Marginal of y is the distribution of each co-ordinate. Here y_i is gaussian distributed with mean (Uy)_i and variance (Cy)_{ii}.
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