Variance of a summation of Gaussians

In summary, the given equation shows that the variance of a sum of Gaussian random variables can be calculated by taking the sum of their individual variances. This is due to the fact that Gaussian random variables have a mean of 0 and a variance of 1, making the calculation simpler. The two facts used in this equation are also common and easy to prove.
  • #1
SeriousNoob
12
0

Homework Statement


I am trying to follow a step in the textbook but I don't understand.

[itex]var\left(\frac{1}{N}\sum_{n=0}^{N-1}w[n]\right)\\
=\frac{1}{N^2}\sum_{n=0}^{N-1}var(w[n])[/itex]
where [itex]w[n][/itex] is a Gaussian random variable with mean = 0 and variance = 1

Homework Equations



[itex]Var(X) = \operatorname{E}\left[X^2 \right] - (\operatorname{E}[X])^2.
[/itex]

The Attempt at a Solution


The mean is 0 because a summation of Gaussian is Gaussian.
But squaring the whole expression doesn't seem right as there seems to be a trick used to go from line 1 to 2.
 
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  • #2
Two facts are being used here:

1. If X is a random variable and c is a constant, then [itex]\text{var}(cX) = c^2\text{var}(X)[/itex].
2. If X and Y are uncorrelated random variables, then [itex]\text{var}(X + Y) = \text{var}(X) + \text{var}(Y)[/itex]. From this, it's an easy induction to handle the sum of N uncorrelated random variables.

Both of these facts are straightforward to prove and should be found in any probability book.
 
  • #3
Thanks a lot. Haven't touched random variables for a while and the summation threw me off.

The proofs for those facts are indeed very straightforward.
 

What is the variance of a summation of Gaussians?

The variance of a summation of Gaussians is the sum of the variances of each individual Gaussian. This means that as more Gaussians are added together, the overall variance will increase.

How is the variance of a summation of Gaussians calculated?

The variance of a summation of Gaussians can be calculated using the formula Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y), where Var(X) and Var(Y) are the variances of each individual Gaussian and Cov(X,Y) is the covariance between them.

What does the variance of a summation of Gaussians tell us?

The variance of a summation of Gaussians tells us the spread or dispersion of the summed distribution. This can be useful in understanding the variability of a data set or in making predictions about future outcomes.

Is the variance of a summation of Gaussians affected by the number of Gaussians added?

Yes, the variance of a summation of Gaussians is directly affected by the number of Gaussians added. As more Gaussians are added, the overall variance will increase, resulting in a wider and more dispersed distribution.

What is the relationship between the variance of a summation of Gaussians and the standard deviation?

The standard deviation is the square root of the variance, so they are directly related. As the variance of a summation of Gaussians increases, the standard deviation will also increase, indicating a wider spread of the distribution.

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