Under the polyploid model for two loci, consider the map (, r) (q00, q11) q00 =

Under the polyploid model for two loci, consider the map

(θ, r) → (q00, q11)

q00 = Pr(X1 = 0, X2 = 0)

q11 = Pr(X1 = 1, X2 = 1).

Show that this map from {(θ, r) : θ ∈ [0, 1], r ∈ (0, 1)} is one to one and onto the region Q = {(q00, q11) : q00 ∈ (0, 1), q11 ∈ (0, 1), q00q11 ≥ q2 01}, where q01 = Pr(X1 = 0, X2 = 1). Prove that θ = 0 if and only if q00 + q11 = 1, and θ = 1 if and only if q00q11 = q201.

The upper boundary of Q is formed by the line q00 + q11 = 1 and the lower boundary by the curve q00q11 = q201.

Prove that the curve is generated by the function q11 =1+ q00 − 2

√q00.