Calculating Sum of Three Correlated Gaussian Distributions

In summary, if you want to find the mean of two distributions with different correlations, you can simply add the two initial means together. Then, you use the variance to find the new stdev.
  • #1
mpoirier
3
0
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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
My bad, I didn't catch the entire problem. :frown:
 
  • #5
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.
 

What is a Gaussian distribution?

A Gaussian distribution, also known as a normal distribution, is a type of probability distribution that is commonly used to model natural phenomena. It is a symmetrical bell-shaped curve that represents the distribution of a continuous random variable.

Why do we add Gaussian distributions?

Adding Gaussian distributions is a common practice in statistics and data analysis. It allows us to combine multiple distributions to better understand the overall distribution of a data set or to make predictions about future outcomes.

How do you add two Gaussian distributions?

To add two Gaussian distributions, you can simply add their means and standard deviations. This is known as the sum of independent normal random variables. For example, if you have two distributions with means of 5 and 10, and standard deviations of 2 and 3, respectively, the resulting distribution will have a mean of 15 and a standard deviation of √(2²+3²) = √13 ≈ 3.61.

What is the Central Limit Theorem and how does it relate to adding Gaussian distributions?

The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables will tend towards a normal distribution, regardless of the underlying distribution of the individual variables. This means that when we add multiple Gaussian distributions, the resulting distribution will also be Gaussian, which makes it a powerful tool for data analysis.

What are some real-world applications of adding Gaussian distributions?

Adding Gaussian distributions is commonly used in fields such as finance, economics, and engineering to model and predict outcomes based on multiple variables. It is also used in machine learning and artificial intelligence for tasks such as image and speech recognition. Additionally, it is used in quality control and process improvement to analyze and improve production processes.

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