So let's say that we have two Gaussian distributions:
Then the sum distribution is simply:
Where for the parameters holds:
In the last formula rho is the correlation coefficient, with value defined as:
Jun 25, 2011
A Gaussian is a Gaussian
Interestingly a sum of Gaussian distributions is again Gaussian distribution. It didn't came as a big surprise to me, but my thesis adviser was truly amazed ... strange. Furthermore it holds true for both uncorrelated, and correlated statistics, and the rule for addition is impressively simple.
So let's say that we have two Gaussian distributions:
 = \frac{1}{\sqrt{2\pi\sigma^{2}_{X}}}\exp\left[-\frac{(t-\mu_{X})^{2}}{2\sigma_{X}^{2}} \right ] "f_{X}(t) = \frac{1}{\sqrt{2\pi\sigma^{2}_{X}}}\exp\left[-\frac{(t-\mu_{X})^{2}}{2\sigma_{X}^{2}} \right ]")
 = \frac{1}{\sqrt{2\pi\sigma^{2}_{Y}}}\exp\left[-\frac{(t-\mu_{Y})^{2}}{2\sigma_{Y}^{2}} \right ] "f_{Y}(t) = \frac{1}{\sqrt{2\pi\sigma^{2}_{Y}}}\exp\left[-\frac{(t-\mu_{Y})^{2}}{2\sigma_{Y}^{2}} \right ]")
Then the sum distribution is simply:
 = \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(t-\mu)^{2}}{2\sigma^{2}} \right ] "f_{X+Y}(t) = \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(t-\mu)^{2}}{2\sigma^{2}} \right ]")
Where for the parameters holds:
In the last formula rho is the correlation coefficient, with value defined as:
}{\sigma_{X}\sigma_{Y}} "\rho=\frac{cov(X,Y)}{\sigma_{X}\sigma_{Y}}")
So let's say that we have two Gaussian distributions:
Then the sum distribution is simply:
Where for the parameters holds:
In the last formula rho is the correlation coefficient, with value defined as:
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