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What is this distribution?

  1. Aug 19, 2011 #1
    Hello everyone (first post here :)),

    I have a question regarding Probability. I calculated the distribution of a square of a sum of two Gaussian variables and I have got this expression (I attached a link to it). (Gaus1+Gaus2)²
    It is like a normal Gaussian distribution (of x) only divided by the square root of the argument x.

    My question is: Does this distribution have a name? I want to check its properties on wiki but I don't know its name.

    Thanks a lot!

    P.S. (if the image linking does not work just copy this link into your browser)
  2. jcsd
  3. Aug 19, 2011 #2

    Stephen Tashi

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    What you calculated doesn't look like a probability distribution. Are you saying that the integral of that function with respect to r, from zero to infinity is 1?

    The sum of two independent gaussian random variables is another gaussian random variable. So I think you are asking for the distribution for the square of a single gaussian random variable, which is a chi square distribution with one degree of freedom.
  4. Aug 19, 2011 #3
    Well I think the variable r should go from 0 to infinity and it does sum up to 1 (since r is actually X² so it should not be negative).
    I have used the formula that is to be found on

    page 303/517 (I'm refering to pages of pdf document, not the pages in book)
    there is an expression for calculation of square of a RV. I plugged in the expression for X which is the sum of 2 Gaussians et voilà.

    I think it is fine since it is in accordance with some results I found in some scientific papers, but then again I might be wrong. Probability is not actually my field :)
  5. Aug 19, 2011 #4

    Stephen Tashi

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    If X and Y are the two unit normal random variables, you say in the original post that your want the distribution of [itex] (X + Y)^2 [/itex] but on page 303 of that document you seem to have used the formula for the Rayleigh distribution, which is the distribution for [itex] (X^2 + Y^2) ^ {\frac{1}{2}} [/itex].
  6. Aug 19, 2011 #5
    Actually when you say that the square of Gaussians gives a chi square it is true if the Gaussians are standard normal (zero mean variance 1), but if they have variance different than 1 then I don't know if this thing holds. Anyway if you take the thing that I derived and put variance equal to one I think you will get Chi square.

    Maybe I should rephrase my Q:
    What is the distribution (of a sum) of squares of non-standard (variance not equal 1) Gaussian variables?
    Or maybe the answer I already have here:
    Say I have Gaussian RV N~(0, sigma).
    If I want the distribution of Gaussian²:
    find the distribution of Gaussian²/sigma (it will be Chi², since it's standard Gaussian - its been normalized)
    its statistic properties are (mean, variance)
    so statistic properties of the wanted (Gaussian²) are (mean*sigma, variance*sigma²)

    And about the derivation itself, starting expression is actually to be found under Theorem 5.1 (page 219/518). Sorry.
  7. Aug 19, 2011 #6
    wow its nice u can insert equations here!!! :)
  8. Aug 19, 2011 #7

    Stephen Tashi

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    You aren't being clear. Are your asking several different questions?

    Your original post asks about summing gaussian variables and then squaring the sum. I'll assume we aren't dealing with that anymore.

    That's one question. I don't know if such a distribution has a nice name or form.
    A related discussion is: https://www.physicsforums.com/showthread.php?t=507750

    That's a different question and I agree you can use a change of variable to transform the distribution of the square of a non-mean-zero non-unit-variance normal random variable to a chi square distribution.
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