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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 $\mu \sim N(0,1)$.

  1. Find $E[\mu|X]$ and $E[(\mu - E[\mu|X])^2 |X]$.
  2. Find the precision of the normal distribution.
  3. 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!

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    I think I know what to do first. I have to find the posterior distribution. Then based off that I can find the mean and variance. Then I can answer number 1. Sorry for wasting anyones time. I just didn't see what to do until now.2011-07-17
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    Betty, you can just answer your own question and tick of the solution as the correct one, so as to not leave this question unanswered.2011-07-17
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    Hmm -- bad coincidence -- I'd just finished answering it :-)2011-07-17

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