What are you going to learn?
Content
Extend and formalize knowledge of the theory of probability and random variables. Compute conditional probabilities directly and using Baye’s theorem and check for independence of events. Perform probability calculations relating to probability distributions for discrete random variables. Perform probability calculations relating to probability density functions for continuous random variables. Compute Mathematical Expectation and variance. Apply various distributions to solve real life problems.
Chapter 1. Combinational Analysis
Principles of Counting
Permutattions
Combinations
Introduction
Sample Space and Events
Axioms of Probability
Probability as a Continuous Set Function
Chapter 2. Probability
Chapter 3. Conditional Probability and Independence
Introduction
Conditional Probabilities
Baye's Formula
Independent Events
Chapter 4. Random Variables
Introduction
Random Variables
Distribution Function
Density Function
Variance
The Uniform Density Random Variable
The Bernoulli and Binomial Random Variable
The Poisson Random Variable
Conditional Distribution/Density Functions
Operation with Random Variables
Chapter 5. Operations on One Random Variable
Introduction
Expectation
Moments
Moment Generating Functions
Transformations of Random Variables
Chapter 6. Multiple Random Variables
Introduction
Joint Distribution Functions
Conditional Distribution and Density
Statistical Independence
Sum of Independent Random Variables
Conditional Dstribution Functions
Chapter 7. Operation on Multiple Random Variables
Introduction
Expected Value of a Function of Random Variables
Joint Characteristic Functions
Jointly Gaussian Random Variables
Transformations of Multiple Random Variables
Linear Transformations of Gaussian Random Variables
Complex Random Variables
Chapter 8. Limit Theorems
Introduction
Chebyshev's Inequality
The Central Limit Theorem
The Strong Law of Large Numbers
Other Inequalities
Chapter 9. Numerical Simulations
Introduction
General Techniques
Simulating Discrete Distributions
Variance Reduction Techniques
Bibliography
Sheldom, R. A. First Course in Probability. 9th ed., Pearson, New York, 2010.
Feller, William . An Introduction to Probability Theory and Its Applications, Vol. 1. 3d ed. John Wiley & Sons, New York, 1968.
