Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By chiro

Math Help - proof for uncorrelated variables X and Y

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    NY, NY
    Posts
    2

    proof for uncorrelated variables X and Y

    Hello .

    Quick question, how can I show that, for uncorrelated variables X and Y, p(X+Y, X-Y) = (Var(X)-Var(Y)/(Var(X)+Var(Y)) ?
    I know how to show that Cov(X+Y, X-Y) = Var(X) - Var(Y).
    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,165
    Thanks
    762

    Re: proof for uncorrelated variables X and Y

    Hey shaunoflobe.

    Recall that Covariance is distributive in that Cov(A+B,C+D) = Cov(A,C) + Cov(A,D) + Cov(B,C) + Cov(B,D)

    Also recall that Cov(X,Y) = E[XY] - E[X]E[Y]. If two random variables are independent, then P(A = a, B = b) = P(A = a)*P(B = b). If you can show Cov(X,Y) = 0 for independent random variables (Hint: What will E[XY] be for this to happen), then you can use Cov(X,X) = Var[X] and collect things together.
    Thanks from shaunoflobe
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2012
    From
    NY, NY
    Posts
    2

    Re: proof for uncorrelated variables X and Y

    Hi Chiro,

    Thanks for the reply,

    Yes I know how to show that Cov(X,Y) = 0:
    As you showed above, if two variables are independent then E(XY) = E(X)E(Y). Therefore, Cov(X,Y) = E(XY) - E(X)(EY) = E(X)E(Y) - E(X)E(Y) = 0.

    and Cov(X,X) = E(X^2) - E(X)^2 = Var(X). Just not sure how I can "collect things together", i.e state the proof. thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,165
    Thanks
    762

    Re: proof for uncorrelated variables X and Y

    Cov(X+Y,X-Y) = Cov(X,X) - Cov(X,Y) + Cov(X,Y) - Cov(Y,Y) = Cov(X,X) - Cov(Y,Y).

    I didn't do the expansion and assumed you may get a Cov(X,Y) term but since you don't, you don't have to worry about it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 15th 2012, 06:53 AM
  2. Replies: 0
    Last Post: October 15th 2012, 06:09 AM
  3. polynomial of two variables proof
    Posted in the Calculus Forum
    Replies: 0
    Last Post: September 7th 2010, 01:46 AM
  4. Uncorrelated, orthogonal and independent???
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: June 10th 2010, 12:16 PM
  5. Replies: 5
    Last Post: October 16th 2008, 02:52 PM

Search Tags


/mathhelpforum @mathhelpforum