#note Population Genetics A Concise Guide (15)

This is a note for the book Population Genetics: A Concise Guide (Second Edition).

Page 28

Problem 2.6 g is almost the same as the homozygosity of the population, G. Suppose we were to define the homozygosity of a population as the probability that two alleles chosen at random from the population with replacement are identical by state. Show that this is equivalent to the definition given in Equation 1.3. Next, show that:

    \[G=\frac{1}{2N}+(1-\frac{1}{2N})g\]

Use this to justify the claim that G and g are “almost the same”. It should be clear that we could have used the term heterozygosity everywhere that we used H without being seriously misled.

Note:

G is defined as the homozygosity of the locus. The total frequency of homozygotes is given by (Equation 1.3):

    \[G=\sum_{i=1}^{k}p_{i}^{2}\]

From another point of view, there are two possibilities for two alleles are identical by state:

1. The two alleles do share an ancestor allele in the previous generation, which is 1/2N.

2. The two alleles do not share an ancestor allele, but their two ancestor alleles are identical by state, which is (1-1/2N)g.

So combine these two possibilities we get:

    \[G=\frac{1}{2N}+(1-\frac{1}{2N})g\]

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