# [R] Overlapping distributions (populations) - assigning an individual to a population?

Wed Apr 9 02:20:04 CEST 2008

```Rolf,

On Wed, 2008-04-09 at 10:57 +1200, Rolf Turner wrote:
> 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)

Thanks for the quick response but I think I need to put some numbers on
this so I can see what you mean.  Say I have two pops with individual
values:

1 2 3 4 5

3 4 5 6 7

and a new individual with value 5 - what is the likelihood of assignment
to each of the populations?

BTW, I say populations, but to keep it simple I didn't go into more
detail - there is no physical overlap in space or time of the
populations/distributions - so there are no gradients from interbreeding
of sub-populations or anything like that.

Regards,

Phil.
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