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Homework Help: Proving that the Guassian Distribution integral converges to 1

  1. Nov 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that the integral of the Guassian Distribution converges to 1:

    [tex]\int_{- \infty}^{\infty} \frac{1}{\sigma \sqrt{2 \pi}} e^{- \frac{(x- \mu )^2}{2 \sigma ^2}} dx = 1[/tex]

    2. Relevant equations


    3. The attempt at a solution

    So I get that I can pull the constants out for the first step, but I have no clue where to go from there. I looked online for a solution and read about having to convert it to polar or something to solve it so now I am really confused. I thought it might work with a simple substitution, one that I just can't think of.

    Any help greatly appreciated.
    Last edited: Nov 17, 2009
  2. jcsd
  3. Nov 17, 2009 #2
    There's another double integral trick you can do, which is a little bit easier. When you have

    [tex]I = \int dx e^{-x^2} [/tex]
    [tex] I^2 = \int dx \int dy e^{-x^2 - y^2} [/tex]

    do a substitution y = x s, dy = x ds.
  4. Nov 17, 2009 #3
    So basically, it ends up like:

    I^2 = \frac{1}{\sigma \sqrt{\pi}} \int dx \int e^{- \frac{(x- \mu )^2}{2 \sigma ^2} - y^2} dy

    But now I'm having trouble with the substitution. Would my substitution look like this?

    y = \frac{(x- \mu )}{2 \sigma} s

    dy = \frac{(x- \mu )}{2 \sigma} ds

    Then I get this:

    I^2 = \frac{1}{\sigma \sqrt{\pi}} \int dx \int \frac{(x- \mu )}{2 \sigma} e^{- \frac{(x- \mu )^2}{2 \sigma ^2} - (\frac{(x- \mu )}{2 \sigma} s)^2} ds

    I feel like I'm doing something horribly wrong and making it more complicated, pardon my ignorance. Thanks for your help so far though, I appreciate it.
  5. Nov 17, 2009 #4
    To make it work since the constant is

    [tex]\frac{1}{\sigma \sqrt{2 \pi}}[/tex]

    I need the integral to equal

    [tex]\sigma \sqrt{2 \pi}[/tex]

    for it to equal 1
  6. Nov 17, 2009 #5
    Get rid of all the constants first. Notice that you can shift x-mu to x since you're integrating over the entire real line, and then do a substitution to make exponential of the form e^(-x^2).
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