Let \(X_{0}, X_{1}\) be given matrices of order \(100 \times 5\) and \(110 \times 5\) such that \(X_{0}^{\prime} X_{0}=\) \(X_{1}^{\prime} X_{1}=F\), and let \(P_{\beta}\) be the Gaussian mixture model

\[
P_{\beta}=\frac{1}{2} N_{100}\left(X_{0} \beta, I_{100}ight)+\frac{1}{2} N_{110}\left(X_{1} \beta, I_{110}ight)
\]

indexed by \(\beta \in \mathbb{R}^{5}\). For any sample point, either \(y \in \mathbb{R}^{100}\) or \(y \in \mathbb{R}^{110}\), show that the likelihood ratio is

\[
\frac{p_{\beta}(y)}{p_{0}(y)}=\exp \left(y^{\prime} X \beta-\beta^{\prime} F \beta / 2ight)
\]

Deduce that the vector \(X^{\prime} y\) is minimal sufficient, and that the sample size \(n(y)\) is not a component of the sufficient statistic.