### Analytic approach

The interest is in inference about given .

With prior

 (24)

, the sample mean, is a sufficient statistic for , as the likelihood, , depends only on and (fixed), through the sampling distribution of which is

 (25) (26)

so the posterior for is

 (27) (28)

So, by inspection

 (29)

is a random sample of size from where is known and the prior for is . The posterior is where

 (30)

and

 (31)

The prior for is and so posterior belief about is as if had a distribution.

danny 2009-07-23