Balding–Nichols model

Short description: Model in population genetics

Balding-Nichols

Probability density function
Cumulative distribution function
Parameters	$0<F<1$(real) $0<p<1$ (real) For ease of notation, let $\alpha =\frac{1-F}{F}p$, and $\beta =\frac{1-F}{F}(1-p)$
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	$1+\sum _{k=1}^{\mathrm{\infty }}\left(\prod _{r=0}^{k-1}\frac{\alpha +r}{\frac{1-F}{F}+r}\right)\frac{t^k}{k!}$
CF	${}_1{F}_1(\alpha ;\alpha +\beta ;it)$

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

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

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.