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Math Help - Covariance question

  1. #1
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    Covariance question

    If Z₁ is Normal (μ₁,σ), Z₂ is Normal (μ₂,σ) and Cov(Z₁,Z₂)=ρσ
    For (X₁,X₂)=(Z₁-Z₂,Z₁+Z₂)
    How can i find the corresponding covariance matrix?
    I know the mean vector is

    μ₁-μ₂
    μ₁+μ₂

    and with Cov(Z₁,Z₂)=ρσ it is possible to find the covariance matrix for Z₁ and Z₂ but i am not sure how to find covariance matrix for X₁,X₂.
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  2. #2
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    Re: Covariance question

    Let C denote the covariance matrix of the random vector Z = (Z_{1},Z_{2}) \ . The random vector X is a linear transformation of the random vector Z.

    X = AZ\ ,

    where

    A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \ .

    The covariance matrix of X is similar to the covariance matrix of Z, i.e.,

    \text{Cov}(X) = ACA^{T}\ ,

    with A^{T} denoting the transpose of the matrix A.
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