I'm confusing myself with this question.
Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use Bayesian inference to update our belief.
Suppose further we have a prior distribution of $\mu$, also Normal, i.e $\mu \sim \mathcal{N}(\mu_0, \sigma_0^2)$, the observation is $x_0$.
Then what is the conditional expectation $\mathbb{E}(X|x_0)$ ? Would you please show me the derivation steps?
I know how to get the posterior distribution on $\mu$ when assuming $\sigma_0^2$ is fixed, i.e
\mu \sim \mathcal{N}(\mu'_0, \sigma_0^2), \;\;\; \mu'_0 = \frac{\sigma_0^2 x_0 + \sigma^2 \mu_0}{\sigma_0^2 + \sigma^2}
Many thanks!