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Integration to find moments

  1. Jun 3, 2009 #1

    I'm trying to use moments to find the mean of a pdf.

    Here is the pdf:

    [tex]f(x|\theta) = 2 \theta^{-2}x^3 exp(\frac{-x^2}{\theta})[/tex]

    I'm not really sure where to start. I can multiply the pdf by X and then integrate with respect to X, but it gives me the wrong answer.

    Any ideas?

  2. jcsd
  3. Jun 3, 2009 #2


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    A wrong answer is a reason to be alert. I'd check the limits of integration and make sure that they are correct in the sense that f(x) > 0 only between those limits.
  4. Jun 3, 2009 #3
    Basically this is what I've got.

    [tex]\int_0^{inf} 2 \theta^{-2}x^{3+2m} dx[/tex]

    Using [tex]y=x^2 / \theta[/tex], if I rearrange this I get somehow:

    [tex]\int_0^{inf} \theta^{m}y^{m+1} dy[/tex]

    Does anyone know where the final x in [tex]x^{3+2m}[/tex] disappears to?
  5. Jun 3, 2009 #4
    You've forgotten about the exponential term in your distribution function.
    [tex]I(k) = \int_0^{\infty} x^k f(x) dx = \int_0^{\infty} 2 \theta^{-2} x^{3+k} e^{{-x^2}/{\theta}} dx[/tex]

    This is a Gaussian integral. See this article down where it says "The general class of integrals of the form..." (equation 9).
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