R help - Apr 2008 - Overlapping distributions (populations) - assigning an individual to a population?

People,

Say a particular measure of an attribute for individuals in different
populations gives a set of overlapping normal distributions (one
distribution per population).  If I then measure this attribute in a new
individual - how do I assess the likelihood of this new individual
belonging to each of the different populations?

Thanks,

Phil.
-- 
Philip Rhoades

Pricom Pty Limited  (ACN 003 252 275  ABN 91 003 252 275)
GPO Box 3411
Sydney NSW      2001
Australia
Fax:     +61:(0)2-8221-9599
E-mail:  phil at pricom.com.au

On 9/04/2008, at 10:30 AM, Phil Rhoades wrote:
> People,
>
> Say a particular measure of an attribute for individuals in different
> populations gives a set of overlapping normal distributions (one
> distribution per population).  If I then measure this attribute in  
> a new
> individual - how do I assess the likelihood of this new individual
> belonging to each of the different populations?
You have a mixture of distributions.  Let the density be

                 k
	f(x) = SUM lambda_i * f_i(x)
                i=1

where the f_i(x) are the densities for the individual components in  
the mixture,
and the lambda_i are the mixing probabilities.

The probability that an individual with observation x is from  
component i is

            lambda_i * f_i(x)
            -----------------
                   f(x)


		cheers,

			Rolf Turner

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