1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.

  2. jcsd
  3. Dec 11, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.,
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook