1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

    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.

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Variance summation Gaussians Date
Summations involving functions Mar 6, 2018
How does one find sample size without a given variance? Nov 7, 2017
Probability , expectation, variance, cross-term vani May 11, 2017