Results 1 to 2 of 2

Math Help - Help with bivariate normal dist question

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    Help with bivariate normal dist question

    Q: Consider U_{1}=Y_{1}+Y_{2} and U_{2}=Y_{1}-Y_{2}. Assume U_{1} and U_{2} have a bivariate normal distribution. Show that U_{1} and U_{2} are independent.

    My work: Suppose U_{1}=Y_{1}+Y_{2} and U_{2}=Y_{1}-Y_{2} have a bivariate normal distribution. To show U_{1} and U_{2} are independent we must show Cov(U_{1},U_{2})=\sum_{i}\sum_{j}\\a_{i}b_{j}Cov(U  _{1},U_{2})=0.

    I am having trouble with the latter computation:

    Cov(U_{1},U_{2})=\sum_{i}\sum_{j}\\a_{i}b_{j}Cov(U  _{1},U_{2})<br />
=a_{1}b_{1}Cov(Y_{1},Y_{1})+
    a_{1}b_{2}Cov(Y_{1},Y_{2})+<br />
a_{2}b_{2}Cov(Y_{2},Y_{2})

    How do I treat Cov(Y_{1},Y_{1}) given U_{1}=Y_{1}+Y_{2} and U_{2}=Y_{1}-Y_{2}?

    All in all, I'm pretty stuck. Any help would be greatly appreciated.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    Cov(Y_{1},Y_{1})=V(Y_1)

    you can prove that via the definition or the short cut formula

    Use Cov(X,Y)=E(XY)-E(X)E(Y) and replace X with Y.

    Cov(Y,Y)=E(YY)-E(Y)E(Y)=E(Y^2)-(E(Y))^2=V(Y)
    Last edited by matheagle; November 23rd 2009 at 11:56 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. the normal dist
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: October 27th 2010, 07:18 PM
  2. Normal / Bivariate Normal distribution
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: May 24th 2010, 03:42 AM
  3. conditional probability with bivariate normal dist. help
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: November 21st 2009, 08:09 PM
  4. Replies: 5
    Last Post: November 12th 2009, 03:03 AM
  5. Normal Dist Prob
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: October 4th 2007, 05:25 AM

Search Tags


/mathhelpforum @mathhelpforum