Integration to find moments

**roadworx** · Jun3-09, 05:57 PM

Hi,

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

Here is the pdf:

$LaTeX Code: f(x|\\theta) = 2 \\theta^{-2}x^3 exp(\\frac{-x^2}{\\theta})$

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?

Thanks.

**EnumaElish** · Jun3-09, 06:47 PM

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.

**roadworx** · Jun3-09, 11:10 PM

Originally Posted by EnumaElish

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.

Basically this is what I've got.

$LaTeX Code: \\int_0^{inf} 2 \\theta^{-2}x^{3+2m} dx$

Using $LaTeX Code: y=x^2 / \\theta$ , if I rearrange this I get somehow:

$LaTeX Code: \\int_0^{inf} \\theta^{m}y^{m+1} dy$

Does anyone know where the final x in $LaTeX Code: x^{3+2m}$ disappears to?

**mXSCNT** · Jun4-09, 12:08 AM

You've forgotten about the exponential term in your distribution function.

Originally Posted by roadworx

Here is the pdf:

$LaTeX Code: f(x|\\theta) = 2 \\theta^{-2}x^3 e^{{-x^2}/{\\theta}}$

$LaTeX Code: I(k) = \\int_0^{\\infty} x^k f(x) dx = \\int_0^{\\infty} 2 \\theta^{-2} x^{3+k} e^{{-x^2}/{\\theta}} dx$

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