What is the Gaussian Integral for Moments?

[roadworx]

Hi,

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?

Thanks.

 

[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.

 

[roadworx]

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.

[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?

 

[mXSCNT]

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

Here is the pdf:

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

[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).