 Solution

Expanding the exponent givesThe Fisher-Neyman factorization theorem therefore shows that $T=X_1+X_2$ is sufficient.

It is also minimal sufficient. For two sample points $x,y\in {\mathbb{R}}^2$,where $C$ is independent of $\mu$. This ratio is independent of $\mu$ exactly when $T(x)=T(y)$. The likelihood-ratio criterion for minimal sufficiency applies, proving the claim and the general normal sample sum with known variance result in this case.