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Q on Bivariate normal

  1. Dec 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Z is a 2x1 multivariate gaussian random vector, where [tex] Z = (X Y)^t [/tex], X,Y are real numbers, with mean zero and covariance matrix

    [tex] \Gamma[/tex]
    which is a 2x2 matrix whose entries are
    [tex] \Gamma_{1,1} = 1[/tex]
    [tex] \Gamma_{1,2} = \alpha [/tex]
    [tex] \Gamma_{2,1} = \alpha [/tex]
    [tex] \Gamma_{2,2} = 1 [/tex]

    with [tex] | \alpha | < 1 [/tex]

    a) Find the joint distribution of [tex] W_1 = X [/tex] and
    [tex] W_2 = X+Y[/tex]

    b) Find the conditional pdf of X+Y given X.




    3. The attempt at a solution

    I think I want to do a linear transformation to get a) but not sure how to attack the problem. Any help/references would be greatly appreciated. This is easy for independent gaussian variables but this is not the case here.

    Thanks.
     
  2. jcsd
  3. Dec 11, 2012 #2

    Ray Vickson

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    Just use the bivariate moment-generating function
    [tex] M(w_1,w_2) = E \exp(w_1 W_1 + w_2 W_2) = \int \int e^{w_1 v_1 + w_2 v_2} f_{W_1 W_2}(v_1,v_2) \: dv_1 \: dv_2.[/tex] See, eg.,
    http://www.public.iastate.edu/~maitra/stat501/lectures/MultivariateNormalDistribution-II.pdf
     
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