# Variance of a summation of Gaussians

## Homework Statement

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

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

## Homework Equations

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

## 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.

Last edited:

jbunniii
Homework Helper
Gold Member
Two facts are being used here:

1. If X is a random variable and c is a constant, then $\text{var}(cX) = c^2\text{var}(X)$.
2. If X and Y are uncorrelated random variables, then $\text{var}(X + Y) = \text{var}(X) + \text{var}(Y)$. 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.

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.