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Subject: Re: Central Moment for a Gaussian Mixture model
Date: Mon, 18 Jul 2011 21:34:09 +0000 (UTC)
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"kumar vishwajeet" wrote in message <j0260l$3u7$1@newscl01ah.mathworks.com>...
> I am trying to calculate the third central moment of a gaussian mixture model having 50 components. I want to know whether the following is a correct method to calculate its third central moment.
> \int{(x-mu)^3 P(x)dx} = E[(x-mu)^3]
>                                 = E[x^3] - 3*mu*E[x^2] - mu^3
> Where mu is the resultant mean of the mixture. and \int indicates the integration.
> .......
- - - - - - - - -
Both of these amount to about the same computation, I would think.  To carry out the integration needed in E{x^3}, E{x^2}, and E{x} you would presumably have to do separate integrations on each of the 50 components.

However I disagree with your last step in the first method.  I believe it should read:

= E{x^3} - 3*mu*E{x^2} + 2*mu^3

since 3*mu^2*E{x} = 3*mu^3.

Roger Stafford