Bayesian Distribution of Sample Mean

prior parameters

μ

1

-5

μ

2

5

σ

1

0.2

σ

2

5

observed parameters

x

2.4

s

2.1

M

11

This Demonstration provides Bayesian estimates of the posterior distribution of the mean

μ

and the standard deviation

σ

of a normally distributed random variable

X

. These posterior distributions are based upon observing

M

independent observations of the random variable

X

that have sample mean

x

and sample distribution

s

. Prior knowledge about statistical parameters is an important part of Bayesian statistics. In this case, it is initially assumed that the unknown mean

μ

is uniformly distributed on the interval

μ

1

<μ<

μ

2

and that the unknown standard deviation

σ

is distributed with a Jeffrey's prior distribution on the interval

σ

1

<σ<

σ

2

. Bayes's theorem provides a convenient way of incorporating the prior information and the observed information into a posterior probabilistic characterization of the unknown parameters

μ

and

σ

.