Solution
= Solution
The common distance modulus of calibrator galaxy $g$ cancels between its supernova and mean Cepheid apparent magnitudes. Their independent intrinsic scatters therefore give
$$
\widehat S_2^g\sim N(\Delta M,V_2),
\qquad
\Delta M=M_0^{\mathrm{SN}}-M_0^*,
\qquad
V_2=\sigma_{\mathrm{SN}}^2+\frac{\sigma_*^2}{N_{\mathrm{Ceph}}}.
$$
Thus $\mathbb E\widehat S_2^g=\Delta M$ and $\operatorname{Var}(\widehat S_2^g)=V_2$.