Let $X_1,X_2,...,X_n$ be a random sample of size $n$ from a population $N(\mu, \sigma^2)$ with $\sigma^2=4$. Further assume that $\mu \sim N(0,1)$.
- Find $E[\mu|X]$ and $E[(\mu - E[\mu|X])^2 |X]$.
- Find the precision of the normal distribution.
- Find a 95% credible interval for $\mu$.
So far I understand what to do for 3, and I know that for 2 it is the inverse of the variance. But I cannot figure out 1. I have looked in my book, which is hard to read and I cannot find anything on how to do 1. If I figure out 1, then I can do 2. Please help with 1. Thank you so much!