Discrete Mathematics, Probability Theory and Stochastic Processes
Li Chen, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
香港科技大学

Textbook:
Discrete Mathematics, Probability Theory and Stochastic Processes: For Applications in Engineering and Computer Science
Modeling and Optimization in Science and Technologies
Author(s): Samir Brahim Belhaouari
College of Science and Engineering Hamad Bin Khalifa University Doha, Qatar

This course provides a comprehensive overview of discrete mathematics, probability theory, and stochastic processes, covering a wide range of topics in each area. It is designed to be a self-contained resource for students wishing to improve their understanding of these important mathematical concepts. The coursebook takes a practical approach to the subject matter, providing real-world examples and applications to help students understand how these mathematical concepts are used in various fields, such as computer science, engineering, and finance.

Contents
1 Topics in Mathematics and Applications
1.1 Methods of Proofs
1.1.1 Direct Proofs
1.1.2 Proof by Contradiction
1.1.3 Proof by Mathematical Induction
1.2 Sequences and Series
1.2.1 Sequences
1.2.2 Arithmetic Sequences
1.2.3 Geometric Sequence
1.2.4 Series
1.2.5 Arithmetic Series
1.2.6 Geometric Series
1.2.7 Means
1.2.8 Applications to Finance
1.2.9 The Limiting Sum of a Geometric Series
1.3 Links Forward
1.3.1 Use of Indication
1.3.2 The Harmonic Series
1.3.3 Connection with Integration
1.3.4 The AM–GM Inequality
1.3.5 History and Applications
1.3.6 The Greeks
1.4 Integer Divisibility
1.4.1 Primes
1.4.2 Perfect Numbers
1.4.3 The Division Algorithm
1.4.4 The Fundamental Theorem of Arithmetic
1.4.5 Counting Divisors
1.4.6 Greatest Common Divisor
1.4.7 Euclidean Algorithm
1.4.8 Continued Fractions
1.5 Diophantine Equations
1.5.1 Linear Diophantine Equations
1.6 Binomial Expansion
1.6.1 Combinatorics
1.7 Modular Arithmetic
1.7.1 Applications of Modular Arithmetic
1.7.2 Solving Linear Congruences
1.7.3 Chinese Remainder Theorem
1.7.4 Arithmetic with Large Integers
1.7.5 The Pigeonhole Principle
1.8 Recurrence Relations
1.8.1 Solving First Order Linear Recurrences
1.8.2 Solving Second Order Recurrences
1.8.3 Iteration
1.8.4 Multiple Roots
1.8.5 Inhomogeneous Equations
1.9 Exercises
Reference
2 Probability Theory and Applications
2.1 Sample Space and Events
2.2 Probability
2.2.1 Probability on Finite Sample Spaces
2.2.2 Independent Events
2.2.3 Bayes’ Theorem
2.3 Random Variables
2.3.1 Discrete Probability Distribution
2.3.2 Continuous Probability Distribution
2.4 Joint Distribution
2.5 Expectation
2.5.1 Mean
2.5.2 Median
2.5.3 Variance and Standard Deviation
2.5.4 Covariance
2.5.5 Correlation Coefficient
2.5.6 Conditional Expectation
2.5.7 Moment Generating Function
2.6 Important Discrete Random Variables
2.6.1 Discrete Uniform
2.6.2 Bernoulli
2.6.3 Binomial
2.6.4 Geometric
2.6.5 Negative Binomial
2.6.6 Hypergeometric
2.6.7 Poisson
2.7 Important Continuous Random Variables
2.7.1 Continuous Uniform
2.7.2 Gamma
2.7.3 Exponential
2.7.4 Normal
2.8 Gambler’s Ruin Problem
2.8.1 Becoming Infinitely Rich or Getting Ruined
2.8.2 Random Walk Hitting Probabilities
2.9 Buffon’s Needle Problem
2.10 Exercises
References
3 Stochastic Processes and Applications
3.1 Gaussian Process
3.2 Ergodic Process
3.3 Point Process
3.3.1 Poisson Process
3.4 Martingale Processes
3.5 Brownian Motion
3.6 Stationarity
3.7 Exercises