# updating p.d.f of mean of a Gaussian random variable with observations

Suppose $X$ is a Gaussian random variable with mean $\mu$.

After observing realizations of $X$, say $\{x_1, \dots, x_N\}$, how do I update the p.d.f using Bayesian formula,

$$f^{t+1}=P(\mu|\{x_1, \dots, x_N\})$$

I need the derivation steps for the update. Thanks for your help!

EDIT:

I realized that $\mu$ is the underlying parameter of $X$. Instead of updating $X$ (which is not right) I should update the p.d.f of $\mu$ when predicting.

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Since mean and variance of the sample will give the maximum likelihood estimates for $\mu$ and $\sigma^2$ you need to compute running means and variances. –  Sasha Feb 6 '12 at 23:03
You'd need a prior probability distributin of $\mu$ and $\sigma^2$. –  Michael Hardy Feb 7 '12 at 0:17
@Sasha Thanks. I would have a look. –  shuaiyuancn Feb 7 '12 at 9:41
@MichaelHardy I cannot give that. All I can do (and want to do) is to give a guess in the beginning then revise according to observations. –  shuaiyuancn Feb 7 '12 at 9:45
@MichaelHardy Thanks. I see what you mean. What are suggested to use as p.d.f for $\mu$ and $\sigma^2$ in the beginning? No clue at all. –  shuaiyuancn Feb 7 '12 at 11:20