Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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.
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.

Estimating the mean of normal distribution with known variance.

e are given a sample MATH from MATH , where the $\theta$ 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 $\theta$ is given by the normal distribution MATH with some known parameters $\gamma$ and $\omega$ . We proceed according to the ( Bayesian technique ) with the components MATH and MATH set according to the expressions MATH MATH We drop the normalization constants from our computation and write

MATH (Known Variance1)
We aim to put the term in the brackets $\{...\}$ into the form MATH independent of $\theta$ terms. The independent terms would be dropped from the calculation because they belong to the normalization constant. We would consequently conclude that MATH . MATH We simplify by keeping only the $\theta$ -dependent terms: MATH MATH Hence, MATH is a normal random variable of the form
MATH (Known Variance2)
where MATH , $\xi$ is MATH . We see that the distribution converges around $\bar{X}$ and gradually forgets the parameters of the prior distribution.

Notation. Index. Contents.

Copyright 2007