Calculating Sum of Three Correlated Gaussian Distributions

[mpoirier]

I have three sets of data that I’ve used to create three Gaussian distributions which have different means and standard deviations. The data sets are also correlated as the data is dependent on time. I want to compare the sum of two distributions with the sum of three distributions to find which sets of distribution produce the best stdev as a percentage of the mean.

Now, I think I know how to sum two of the distributions together, but how do I sum the three?

To find the new mean of two distributions, simply add the two initial means together.
To find the new stdev of the two distributions, use the following formulas.
First, find the variance.
(1) Var(X1, X2) = stdev1^2 + stdev2^2 + 2cov(X1, X2)
(2) Correl(X1, X2) = cov(X1, X2) / (stdev1 * stdev2)
And you obtain… (2) into (1)
Var(X1, X2) = stdev1^2 + stdev2^2 + 2 * Correl(X1,X2) * (stdev1 * stdev2)

Then the stdev is simple the square root of the variance.

None of the three samples are completely independent or dependent. In other words, the correlation is not equal to 0 or 1.

Any help would be greatly appreciated.

 

[Hurkyl]

Now, I think I know how to sum two of the distributions together, but how do I sum the three?

The same way you would add three numbers together: you start by adding two of them.

 

[mpoirier]

Sorry, I don't think I explained the question properly.
The problem is that there is a different correlation between each data set.
For example:
Set 1 and Set 2 have a correl of .71
Set 1 and Set 3 have a correl of .80
Set 2 and Set 3 have a correl of .70

So, if I simply solve it out twice, what do I use for the correl between the solution of Set 1/2 and Set 3? That's if I solve it the way you just suggested.

 

[Hurkyl]

My bad, I didn't catch the entire problem.

 

[mpoirier]

Ok, I figured out how to answer my own question. You just plug in all the numbers into the following equation.
Var(X1, X2, X3) = stdev1^2 + stdev2^2 + stdev3^2 + 2 * Correl(X1,X2) * (stdev1 * stdev2) + 2 * Correl(X1,X3) * (stdev1 * stdev3) + 2 * Correl (X2,X3) * (stdev2 * stdev3)

Anyways, thanks Hurkyl for trying to help.