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

I found the multivariate density to be

<br />
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)}<br />

I get for the posterior Bayes:

<br />
\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}<br />

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

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

Thank you!