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.^[1] 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

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

Balding-Nichols

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

where B is the Beta distribution. This distribution has mean p and variance Fp(1 – p).^[2]

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.