Correlation of combined variables

• Aug 6th 2010, 02:54 AM
mpb
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
• Aug 6th 2010, 12:43 PM
SpringFan25
First, note that for any 3 Random variables (X,Y,Z):
Cov(X+Y,Z) = Cov(X,Z) + Cov(Y,Z)

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

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.
• Aug 7th 2010, 05:30 AM
mpb
At last the penny's dropped - thank you very much for your help!