Multiplicative diallelic penetrance ($q_G=p{\theta }^G,G\sim \mathrm{Bin}(2,\varphi )$)

With Hardy-Weinberg equilibrium, let $G$ count copies of the risk allele. The penetrance $p{\theta }^G$ must lie in $[0,1]$ for every possible genotype. Writing $\alpha =1-\varphi +\varphi \theta$ and $\beta =1-\varphi +\varphi {\theta }^2$ gives $\mathbb{E}(q_G)=p{\alpha }^2$ and $\mathbb{E}(q_G^2)=p^2{\beta }^2$.