Inbreeding equations

Derivation of Inbreeding equations

F Inbreeding coefficient = expectation that two alleles are identical by descent (autozygous):
                                                    exact genetic copies of a DNA sequence in common ancestor.

Alleles may be identical by allelic state, and not identical by descent.
    Theory and equations should be applied only to the latter.
          If it is assumed [see note below] that observed allelic variants arise only once,
                identity by allelic state is the same as identity by descent

F is also the proportion of population that is inbred at any locus: the fraction of individuals with two alleles identical by descent.
Then, homozygosity at any locus indicates identity by descent

What is the effect of inbreeding on genotype proportions?

In the absence of inbreeding, expected f(AA) = p²
f(AB) = 2pq
f(BB) = q²
In the presence of inbreeding,

f(AA) = (1 - F)(p²) + (F)(p)(1) = p² - Fp² + Fp = p² + Fp(1 - p) = p² + Fpq
       fraction (1 - F) of population not inbred:
            expected frequency of AA homozygotes among these = p²
       fraction (F) of population inbred:
          fraction p of these individuals have A allele
        If inbred, other allele must also be A, with probability = 1

f(AB) = (1 - F)(2pq) + (F)(0)     = 2pq - 2Fpq      = 2pq (1 - F)
        fraction (1- F) of population not inbred:
     the expected frequency of AB heterozygotes among these is 2pq
fraction (F) of population inbred:
     among these, no heterozygotes, since alleles not identical.

f(BB) = (1 - F)(q²) + (F)(q)(1) = q² - Fq² + Fq = q² + Fpq
Follow same logic for f(AA) above, applied to B allele