Results 1 to 3 of 3

Math Help - Bivariate Normal Distribution: Joint distribution of functions of random variables

  1. #1
    Newbie
    Joined
    Mar 2013
    From
    MNL
    Posts
    2

    Bivariate Normal Distribution: Joint distribution of functions of random variables

    Hi, I need your help with this problem: Suppose (X, Y)' follows a Bivariate Normal Distribution with parameters μ1 ,μ2, σ1^2, σ2^2, and ρ. Let U = X + Y and V = X - Y. Considering that X and Y are not independent random variables, how will I get the joint distribution of U and V? Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,607
    Thanks
    591

    Re: Bivariate Normal Distribution: Joint distribution of functions of random variable

    Hey Mach.

    You will have to use a change of variables in your integral. Have you come across the substitution theorem for multivariable integration?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie ButterflyM's Avatar
    Joined
    Mar 2013
    From
    Cambridge
    Posts
    8

    Re: Bivariate Normal Distribution: Joint distribution of functions of random variable

    Well, I believe you can use their linear combination theorem since they are assumed bivariate normal, that is when we have the scalar product

    U=AX+BY=(A B)'(X Y). In this case, we need A=1, B=1 to have U=X+Y

    The mean of U is thus given by

    \mu_U=(1 1) (\mu_x \mu_y)'=\mu_x+\mu_y

    The variance of U is given by

    \sigma^2_U=(1 1) \Sigma_{X,Y}(1 1)'

    where \Sigma_{X,Y} is the covariance matrix of X and Y.

    Note that the variance matrix Sigma is not just a diagonal matrix like in the independence case. Here, the covariance elements \sigma_{x,y} are assumed to be non zero.

    Therefore, we obtain \sigma^2_U=\sigma^2_X+\sigma^2_Y+2 \sigma_{X,Y} (as expected)

    And since they are bivariate normal by assumption, we obtain

    U \sim N(\mu_x+\mu_y, \sigma^2_X+\sigma^2_Y+2 \sigma_{X,Y})

    Similar for V. In order to get V, replace (1 1) by (1 -1) as then you have V=X-Y. Then redo the matrix calculations for the joint mean and variance.
    Last edited by ButterflyM; March 23rd 2013 at 04:44 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 21st 2013, 02:59 AM
  2. Replies: 3
    Last Post: November 13th 2011, 09:53 AM
  3. Replies: 3
    Last Post: August 13th 2011, 12:10 PM
  4. Replies: 2
    Last Post: October 17th 2010, 07:16 PM
  5. Replies: 0
    Last Post: February 7th 2010, 07:56 PM

Search Tags


/mathhelpforum @mathhelpforum