Let $X_{1}, \ldots, X_{n}$ be i.i.d. random variables from a $N(\theta, 1)$ distribution, $\theta \in \mathbb{R}$, and consider a Bayesian model $\theta \sim N\left(0, v^{2}\right)$ for the unknown parameter, where $v>0$ is a fixed constant.

(a) Derive the posterior distribution $\Pi\left(\cdot \mid X_{1}, \ldots, X_{n}\right)$ of $\theta \mid X_{1}, \ldots, X_{n}$.

(b) Construct a credible set $C_{n} \subset \mathbb{R}$ such that

(i) $\Pi\left(C_{n} \mid X_{1}, \ldots, X_{n}\right)=0.95$ for every $n \in \mathbb{N}$, and

(ii) for any $\theta_{0} \in \mathbb{R}$,

$$P_{\theta_{0}}^{\mathbb{N}}\left(\theta_{0} \in C_{n}\right) \rightarrow 0.95 \quad \text { as } n \rightarrow \infty,$$

where $P_{\theta}^{\mathbb{N}}$ denotes the distribution of the infinite sequence $X_{1}, X_{2}, \ldots$ when drawn independently from a fixed $N(\theta, 1)$ distribution.

[You may use the central limit theorem.]