Let $\left(X_{1}, Y_{1}\right), \ldots,\left(X_{n}, Y_{n}\right)$ be jointly independent and identically distributed with $X_{i} \sim N(0,1)$ and conditional on $X_{i}=x, Y_{i} \sim N(x \theta, 1), i=1,2, \ldots, n$.

(a) Write down the likelihood of the data $\left(X_{1}, Y_{1}\right), \ldots,\left(X_{n}, Y_{n}\right)$, and find the maximum likelihood estimate $\hat{\theta}$ of $\theta$. [You may use properties of conditional probability/expectation without providing a proof.]

(b) Find the Fisher information $I(\theta)$ for a single observation, $\left(X_{1}, Y_{1}\right)$.

(c) Determine the limiting distribution of $\sqrt{n}(\hat{\theta}-\theta)$. [You may use the result on the asymptotic distribution of maximum likelihood estimators, without providing a proof.]

(d) Give an asymptotic confidence interval for $\theta$ with coverage $(1-\alpha)$ using your answers to (b) and (c).

(e) Define the observed Fisher information. Compare the confidence interval in part (d) with an asymptotic confidence interval with coverage $(1-\alpha)$ based on the observed Fisher information.

(f) Determine the exact distribution of $\left(\sum_{i=1}^{n} X_{i}^{2}\right)^{1 / 2}(\hat{\theta}-\theta)$ and find the true coverage probability for the interval in part (e). [Hint. Condition on $X_{1}, X_{2}, \ldots, X_{n}$ and use the following property of conditional expectation: for $U, V$ random vectors, any suitable function $g$, and $x \in \mathbb{R}$,

$$P\{g(U, V) \leqslant x\}=E[P\{g(U, V) \leqslant x \mid V\}] .]$$