Variance of a summation of Gaussians

[SeriousNoob]

Homework Statement

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

var\left(\frac{1}{N}\sum_{n=0}^{N-1}w[n]\right)\\<br /> =\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.<br />

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.

 

[jbunniii]

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.

 

[SeriousNoob]

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.