2020-34.md

course	course_year	question_number	tags	title	year
Statistics	IB	34	IB 2020 Statistics	Paper 1, Section I, $\mathbf{6 H}$	2020

Suppose $X_{1}, \ldots, X_{n}$ are independent with distribution $N(\mu, 1)$. Suppose a prior $\mu \sim N\left(\theta, \tau^{-2}\right)$ is placed on the unknown parameter $\mu$ for some given deterministic $\theta \in \mathbb{R}$ and $\tau>0$. Derive the posterior mean.

Find an expression for the mean squared error of this posterior mean when $\theta=0$.