We are given a simple linear regression model, $y_i$ = $\beta$$x_i$ + $\epsilon$$_i$ and $\epsilon$$_i$ ~ N(0, 1). We are then asked to calculate the likelihood of the model in terms of $\beta$. I was able to solve that.
We are then told the prior for $\beta$ is N(0,100). We are asked to find the posterior distribution of $\beta$ given a dataset. I found:
$P(\beta | x, y)$ $\propto$ $\exp\left[\sum \left(- \frac{{(y_i - \beta x_i)^2}}{2}\right) - \frac{{\beta^2}}{200}\right]$. I know this is going to be a normal distribution. However, I want to simplify it as N($\mu$, $\sigma^2$) but I am not being able to find the values of $\mu$ and $\sigma^2$.
Can someone help me to find the simplified posterior distribution of $\beta$?
Thank you in advance.