## Posterior Bayes Estimator

Hi,

I'm trying to find the posterior Bayes Estimator for a prior $\displaystyle \pi(\theta) = 1, \theta \in R$. The distribution for $\displaystyle X_1, X_2, ..., X_n$ is $\displaystyle N(\theta, \sigma^2)$, with $\displaystyle \theta$ is unknown.

I found the multivariate density to be

$\displaystyle f(\bold X|\theta) = (\frac {1} {2\pi\sigma^2})^{(n/2)} e^{(-\frac {1} {2\sigma^2})(\sum_i{x_i^2}-2\theta\sum_i{x_i}+n\theta^2)}$

I get for the posterior Bayes:

$\displaystyle \pi(\theta|\bold X) = \frac {e^{-\frac {1} {2\sigma^2}(n\theta^2-2\theta\sum_i{x_i})}} {\int {e^{-\frac {1} {2\sigma^2}(n\theta^2-2\theta\sum_i{x_i})}}d\theta}$

I could expand the exponent again to make the denominator look like a normal distribution and then integrate to get it equal to $\displaystyle \frac {1} {2\pi\sigma^2}$ and then have the identical normal density in the numerator. That would then give me $\displaystyle E(theta|\bold X) = \mu$.

Does that look right? Where would the part about the prior being improper come in?

Thank you!