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Variance of a summation of Gaussians

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data
    I am trying to follow a step in the text book 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
    2. Relevant equations

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


    3. 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: Sep 29, 2012
  2. jcsd
  3. Sep 29, 2012 #2

    jbunniii

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    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.
     
  4. Sep 29, 2012 #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.
     
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