Question: The fixation index of a population and the inbreeding coefficient of an individual are two differ…

The fixation index of a population and the inbreeding
coefficient of an individual are two different quantities but they
are closely related to one another.

The fixation index is defined as: F = (H_e −
H_o) / H_e

where H_o is the observed proportion of heterozygotes
in a population, and He is the proportion expected given
Hardy-Weinberg assumptions. It’s a convenient statistic that can be
calculated when genotype proportions have been observed in a sample
from a population.

In class we showed in a simple model that if we know the
inbreeding coefficient f for a class of individuals, and the
population frequencies of the A allele (p) and a allele (q), then
the proportion of AA homozygotes in the inbred individuals will be
p_AA = pf+(1-f)p² . We derived this result
using the rules of conditional probability. We also concluded that
the proportion of Aa heterozygotes among in the inbred individuals
will be p_Aa = 2pq(1-f).

A) Using the principles of conditional probability, show why the
proportion of Aa heterozygotes among the inbred individuals will be
p_Aa = 2pq(1-f). Assume that there are no new mutations.
This implies that two loci that are inherited IBD will be identical
by state.

B) Recognizing that pAa is the observed proportion of
heterozygotes in a population (H_o) and 2pq is the
expected Hardy Weinberg proportion for heterozygotes in the
population (H_e), show algebraically how the inbreeding
coefficient f of a class of inbred individuals is related to the
fixation index F for the same set of individuals. (Hint: attempt to
express in terms of Ho and He).