Balding–Nichols model

Balding-Nichols

Probability density function
Cumulative distribution function
Parameters	$0<F<1$$0<F<1$(real) $0<p<1$$0<p<1$ (real) For ease of notation, let $\alpha ={\tfrac {1-F}{F}}p$$\alpha ={\tfrac {1-F}{F}}p$, and $\beta ={\tfrac {1-F}{F}}(1-p)$$\beta ={\tfrac {1-F}{F}}(1-p)$
Support	$x\in (0;1)\!$$x\in (0;1)\!$
PDF	${\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!$${\frac {x^{{\alpha -1}}(1-x)^{{\beta -1}}}{{\mathrm {B}}(\alpha ,\beta )}}\!$
CDF	$I_{x}(\alpha ,\beta )\!$$I_x(\alpha,\beta)\!$
Mean	$p\!$$p\!$
Median	$I_{0.5}^{-1}(\alpha ,\beta )$$I_{{0.5}}^{{-1}}(\alpha ,\beta )$ no closed form
Mode	${\frac {F-(1-F)p}{3F-1}}$${\frac {F-(1-F)p}{3F-1}}$
Variance	$Fp(1-p)\!$$Fp(1-p)\!$
Skewness	${\frac {2F(1-2p)}{(1+F){\sqrt {F(1-p)p}}}}$${\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!}}$$1+\sum _{{k=1}}^{{\infty }}\left(\prod _{{r=0}}^{{k-1}}{\frac {\alpha +r}{{\frac {1-F}{F}}+r}}\right){\frac {t^{k}}{k!}}$
CF	${}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!$${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$

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

Differential equation

$\left\{F(x-1)xf'(x)+f(x)(F(-p)+3Fx-F+p-x)=0\right\}$$\left\{F(x-1)xf'(x)+f(x)(F(-p)+3Fx-F+p-x)=0\right\}$