Multiplying the Gaussian likelihood and prior and <completing the square> gives
$$
p(\theta\mid y)
\propto
\exp\!\left[-\frac12\left\{
\frac{(y-\theta)^2}{\sigma^2}+\frac{\theta^2}{\tau^2}
\right\}\right].
$$
Therefore <Normal-normal conjugacy> gives
$$
\theta\mid y\sim N(m,v),
\qquad
m=\frac{\tau^2}{\sigma^2+\tau^2}y,
\qquad
v=\frac{\sigma^2\tau^2}{\sigma^2+\tau^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 4 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-219/4/a/ii/solution