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:

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

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]

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.

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

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

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

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

 

[mXSCNT]

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

Here is the pdf:

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

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

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