Hardy-Weinberg Equilibrium. In a certain large population, a gene has
two alleles a and A, with respective proportions θ and 1 − θ. Assume these
same proportions hold for both males and females. Also assume there is no
migration in or out and no selective advantage for either a or A, so these
proportions of alleles are stable in the population over time. Let the genotypes
aa = 1, Aa = 2, and AA = 3 be the states of a process. At step 1, a female
is of genotype aa, so that X1 = 1. At step 2, she selects a mate at random
and produces one or more daughters, of whom the eldest is of genotype X2.
At step 3, this daughter selects a mate at random and produces an eldest
daughter of genotype X3, and so on.
a) The X-process is a Markov chain. Find its transition matrix. For example,
here is the argument that p12 = 1 − θ: A mother of type aa = 1 surely
contributes the allele a to her daughter, and so her mate must contribute
an A-allele in order for the daughter to be of type Aa = 2. Under random
mating, the probability of acquiring an A-allele from the father is 1 − θ.
b) Show that this chain is ergodic. What is the smallest N that gives PN > 0?
c) According to the Hardy-Weinberg Law, this Markov chain has the “equilibrium” (steady-state) distribution σ = [θ2, 2θ(1 − θ), (1 − θ)2]. Verify
that this is true.
d) For θ = 0.2, simulate this chain for m = 50 000 iterations and verify
that the sampling distribution of the simulated states approximates the
Hardy-Weinberg vector.
A-allele upon mating, and the probability of an Aa offspring is 80% = 1−θ. If there
are only 70 AAs among the males, then there must be 20 Aas. The probability that
an Aa mate contributes an A-allele is 1/2, so that the total probability of an Aa
offspring is again 1(0.70) + (1/2)(0.20) = 80% = 1 − θ. Other apportionments of
genotypes AA and Aa among males yield the same result. The first row of the matrix
P is [θ, 1 − θ, 0]; its second row is [θ/2, 1/2, (1 − θ)/2]. (b) For the given σ, show
that σP = σ. (d) Use a program similar to the one in Example 7.1.