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Balding-Nichols
	Probability density function
	Cumulative distribution function
Parameters	(real); (real); For ease of notation, let; , and ;
Support
PDF
CDF
Mean
Median	no closed form
Mode
Variance
Skewness
MGF
CF

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

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).^[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.

References

^ Balding, DJ; Nichols, RA (1995). "A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity". Genetica. 96 (1–2). Springer: 3–12. doi:10.1007/BF01441146. PMID 7607457. S2CID 30680826.
^ Alkes L. Price; Nick J. Patterson; Robert M. Plenge; Michael E. Weinblatt; Nancy A. Shadick; David Reich (2006). "Principal components analysis corrects for stratification in genome-wide association studies" (PDF). Nature Genetics. 38 (8): 904–909. doi:10.1038/ng1847. PMID 16862161. S2CID 8127858. Archived from the original (PDF) on 2008-07-03. Retrieved 2009-02-19.

[1] Balding, DJ; Nichols, RA (1995). "A method for quantifying differentiation between populations at multi-allelic loci and its implications for investigating identity and paternity". Genetica. 96 (1–2). Springer: 3–12. doi:10.1007/BF01441146. PMID 7607457. S2CID 30680826.

[2] Alkes L. Price; Nick J. Patterson; Robert M. Plenge; Michael E. Weinblatt; Nancy A. Shadick; David Reich (2006). "Principal components analysis corrects for stratification in genome-wide association studies" (PDF). Nature Genetics. 38 (8): 904–909. doi:10.1038/ng1847. PMID 16862161. S2CID 8127858. Archived from the original (PDF) on 2008-07-03. Retrieved 2009-02-19.

[1]

[2]

Population genetics
Key concepts	Hardy–Weinberg principle Genetic linkage Identity by descent Linkage disequilibrium Fisher's fundamental theorem Neutral theory Shifting balance theory Price equation Coefficient of inbreeding Coefficient of relationship Selection coefficient Fitness Heritability Population structure Constructive neutral evolution
Selection	Natural Artificial Sexual Ecological
Effects of selection on genomic variation	Genetic hitchhiking Background selection
Genetic drift	Small population size Population bottleneck Founder effect Coalescence Balding–Nichols model
Founders	R. A. Fisher J. B. S. Haldane Sewall Wright
Related topics	Biogeography Evolution Evolutionary game theory Fitness landscape Genetic genealogy Landscape genetics and genomics Microevolution Population genomics Phylogeography Quantitative genetics
Index of evolutionary biology articles

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	${}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!$

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References