For a locus with two alleles, show that the additive genetic variance satisfies 2 a = 2p1p2(1

For a locus with two alleles, show that the additive genetic variance satisfies

σ2 a = 2p1p2(α1 − α2)

2

= 2p1p2[p1(µ11 − µ12) + p2(µ12 − µ22)]2. (6.10)

As a consequence of formula (6.10), σ2 a can be 0 only in the unlikely circumstance that µ12 lies outside the interval with endpoints µ11 and µ22. (Hint: Expand 0 = 2(α1p1 + α2p2)2 and subtract from the expression defining σ2 a.)

Show that the dominance genetic variance satisfies

σ2 d = p2 1p2 2(µ11 − 2µ12 + µ22)

2.

It follows that if either p1 or p2 is small, then σ2 d will tend to be small compared to σ2

a. Hint: Let µ = p2 1µ11+ 2p1p2µ12 +p2 2µ22. Since µ = 0, it follows that

δ11 = µ11 − 2α1 + µ

= p2 2(µ11 − 2µ12 + µ22)

δ12 = −p1p2(µ11 − 2µ12 + µ22)

δ22 = p2 1(µ11 − 2µ12 + µ22).