Estimating the mean of normal distribution with known variance.

e are given a sample
from
,
where the
is an unknown random variable. We will be following the outline of the section
(
Basic idea of Bayesian
analysis
). Our prior knowledge about the random variable
is given by the normal
distribution
with some known parameters
and
.
We proceed according to the (
Bayesian
technique
) with the components
and
set according to the
expressions
We drop the normalization constants from our computation and
write

(Known Variance1)

We aim to put the term in the brackets
into the form
independent
of
terms. The independent terms would be dropped from the calculation because
they belong to the normalization constant. We would consequently conclude that
.
We simplify by keeping only the
-dependent
terms:
Hence,
is a normal random variable of the
form

(Known Variance2)

where
,
is
.
We see that the distribution converges around
and gradually forgets the parameters of the prior distribution.