Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
1. Conditional probability.
A. Definition of conditional probability.
B. A bomb on a plane.
C. Dealing a pair in the "hold' em" poker.
D. Monty-Hall problem.
E. Two headed coin drawn from a bin of fair coins.
F. Randomly unfair coin.
G. Recursive Bayesian calculation.
H. Birthday problem.
I. Backward induction.
J. Conditional expectation. Filtration. Flow of information. Stopping time.
2. Normal distribution.
3. Brownian motion.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Randomly unfair coin.

onsider a machine that makes unfair coins. A coin that comes out of the machine has probability $p$ of head when tossed. The $p$ itself is a random variable sampled from uniform distribution (on [0,1]). We have some coin with unknown $p$ . We toss the coin and obtain head. What is the distribution of the result of the next toss of the same coin?

Let $H_{k}$ denote $H$ (head) at $k$ -th toss. We are interested in MATH . We introduce a uniform mesh MATH on $\left[ 0,1\right] $ and calculate using the formula ( Bayes formula ): MATH According to the formula ( Total probability rule ), MATH MATH MATH MATH MATH MATH Hence, MATH

Notation. Index. Contents.

Copyright 2007