 Solution

DefineThus $F_{+}$ is the equal mixture of $N(0,1)$ and $N(2b_1,1)$, while $F_{-}$ is the equal mixture of $N(0,1)$ and $N(-2b_1,1)$. They have finite variance and everywhere positive densities. Normal symmetry shows that $F_{+}$ is symmetric about $b_1$ and $F_{-}$ about $-b_1$.

For every $t$,where the maximum occurs at $t=b_1$. ConsequentlyThe same argument, shifted and reflected, applies to $F_{-}$. Hence both distributions belong to ${\mathcal{P}}_{\epsilon }^K(\Phi )\cap \mathcal{M}$ and satisfy the required translation relation.