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.

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