In the hypergeometric polygenic model, suppose that one randomly samples each of the n polygenes transmitted to a gamete with replacement rather than without replacement. If j = i is not a descendant of i, and i has parents k and l, then show that this altered model entails E(Xi)=0 Cov(Xi, Xj ) = 1 2 Cov(Xk, Xj ) + 1 2 Cov(Xl, Xj )

Var(Xi)=2n +

1 4



1 − 1 n



Var(Xk) + 1 4



1 − 1 n



Var(Xl)

+

1 2 Cov(Xk, Xl).