I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 2 Classical statistics.
 3 Bayesian statistics.
 A. Basic idea of Bayesian analysis.
 B. Estimating the mean of normal distribution with known variance.
 C. Estimating unknown parameters of normal distribution.
 a. Structure of the model with unknown parameters.
 b. Recursive formula for posterior joint distribution.
 c. Marginal distribution of mean.
 d. Marginal distribution of precision.
 D. Hierarchical analysis of normal model with known variance.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Recursive formula for posterior joint distribution.

roposition

With notation of the previous section the posterior distribution takes the form of the prior distribution: with the parameters given by the expressions where the and are sample's mean and standard deviation ,

Proof.

We replace the dependence on with the dependence on the statistics and : We used the results of the section ( Sufficient statistics for normal sample section ). Therefore, in line with the calculation ( Normal distribution with unknown parameters 1 )-( Normal distribution with unknown parameters 2 ), We extract the term from the above expression: Hence, or and the claim follows.

 Notation. Index. Contents.