 Solution

The parameters are $\mu \in \mathbb{R}$, $|\varphi |<1$ and innovation variance ${\sigma }^2>0$. Stationarity gives $X_1\sim N(\mu ,{\sigma }^2/(1-{\varphi }^2))$, and the Markov property gives $X_t\mid X_{t-1}\sim N(\mu +\varphi (X_{t-1}-\mu ),{\sigma }^2)$. Multiplying the initial density and the transition densities yields the stationary Gaussian AR1 likelihood, with $n=31$:The initial-observation factor depends on all three parameters and must be retained for the full stationary likelihood function. Omitting it gives a conditional likelihood given $X_1$, a different objective.