Results 1 to 3 of 3

Math Help - Correlation of combined variables

  1. #1
    mpb
    mpb is offline
    Newbie
    Joined
    Aug 2010
    Posts
    3

    Correlation of combined variables

    Dear all

    I have got the following problem:

    Given are two correlated, normally distributed variables A and B and corresponding variances and correlation. Both means are zero.

    Now each variable is combined (with different weights a and b) with an additional normally distributed variable C, which has zero mean as well, known variance and is not correlated with A or B. That means:

    A_combined = a*A + (1-a)*C
    B_combined = b*B + (1-b)*C

    Is there any way to analytically derive the correlation between A_combined and B_combined? The solution for equal weights a=b is quite straight forward, but I really got my wires crossed for a<>b...

    Thank you very much for your help!

    Marc
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    First, note that for any 3 Random variables (X,Y,Z):
    Cov(X+Y,Z) = Cov(X,Z) + Cov(Y,Z)

    Since
    Cov(X+Y,Z) = E \left[((X+Y - \mu_x -\mu_y)(Z - \mu_z) \right]
    =E \left[((X - \mu_x)(Z - \mu_z) + (Y - \mu_y)(Z - \mu_z) \right]
    =E \left[((X - \mu_x)(Z - \mu_z)\right] + E \left[(Y - \mu_y)(Z - \mu_z) \right]
    =Cov(X,Z) + Cov(Y,Z)


    Apply that rule twice:


    Cov(aA + (1-a)C,bB + (1-b)C) = Cov(aA,bB) + Cov(aA,(1-b)C) + Cov((1-a)C,bB) + Cov((1-a)C,(1-b)C)
    ...

    You can evaluate each of those terms provided you know that Cov(aX,Y) = aCov(X,Y)

    Once you have the covariance, you should be able to get the correlation.
    Last edited by SpringFan25; August 6th 2010 at 01:53 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    mpb
    mpb is offline
    Newbie
    Joined
    Aug 2010
    Posts
    3
    At last the penny's dropped - thank you very much for your help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: October 17th 2011, 05:23 AM
  2. Correlation with multiple variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: September 8th 2011, 11:41 AM
  3. Correlation of Two Random Variables
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: April 29th 2010, 09:59 AM
  4. Mean and variance of two stochastic variables combined
    Posted in the Advanced Statistics Forum
    Replies: 7
    Last Post: April 10th 2010, 12:26 PM
  5. Covariance and Correlation(continuous random variables)
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: October 22nd 2009, 01:32 AM

Search Tags


/mathhelpforum @mathhelpforum