The random effect model has form: $$S_{ij} = X_{i}^\top \beta + \alpha_i + \epsilon_{ij} $$ where we observe $S_{ij}$, we know $X_{i}$, and $\alpha_{i} \sim N(0, \nu^2), \epsilon_{ij}\sim N(0,\sigma^2)$. Here $i$ is the subject and $\alpha_i$ is the subject intercept, and $i,j$ pair denote the $j-$th observation of the $i-$th subject. The unknown parameters are $\theta := (\beta, \nu, \sigma)$.
Now I would like to answer:
What is the conditional distribution of $\alpha_{i}$ given $S_{ij}, \ \forall i,j.$
I rewrite $\alpha_i = S_{ij} - X_i^\top\beta - \epsilon_{ij}$, but obviously saying $\alpha_i |S_{ij} \sim N(S_{ij} - X_{i}^\top\beta,\sigma^2)$ is wrong since $a_i$ is also a normal variable.
Is there any resource I can see the answer to this?
Edit: I found that this is question 1.4. part 2 in one of Stanford's exercise sheets.
