Balding–Nichols model

In population genetics, the Balding–Nichols model is a statistical description of the allele frequencies in the components of a sub-divided population. With background allele frequency p the allele frequencies, in sub-populations separated by Wright's F_ST F, are distributed according to independent draws from

Balding-Nichols

Probability density function
Cumulative distribution function
Parameters	$0<F<1$(real) $0<p<1$ (real) For ease of notation, let $\alpha ={\tfrac {1-F}{F}}p$, and $\beta ={\tfrac {1-F}{F}}(1-p)$
Support	$x\in (0;1)\!$
PDF	${\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!$
CDF	$I_{x}(\alpha ,\beta )\!$
Mean	$p\!$
Median	$I_{0.5}^{-1}(\alpha ,\beta )$ no closed form
Mode	${\frac {F-(1-F)p}{3F-1}}$
Variance	$Fp(1-p)\!$
Skewness	${\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}$
MGF	$1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{{\frac {1-F}{F}}+r}}\right){\frac {t^{k}}{k!}}$
CF	${}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!$

$B\left({\frac {1-F}{F}}p,{\frac {1-F}{F}}(1-p)\right)$

where B is the Beta distribution. This distribution has mean p and variance Fp(1 – p).

The model is due to David Balding and Richard Nichols and is widely used in the forensic analysis of DNA profiles and in population models for genetic epidemiology.