Probability Theory and Stochastic Processes | Fundamentals of Mathematics

🎯 Series Overview

Probability theory and stochastic processes are the mathematical foundations for uncertainty quantification, process control, and data analysis in materials science. This series covers from the basics of probability variables and distributions to the law of large numbers, central limit theorem, Markov processes, Poisson processes, and stochastic differential equations (SDE), learning theory and Python implementation in pairs. Practical applications including uncertainty modeling in materials processes, quality control, failure prediction, and time series data analysis are also covered.

Learning Path

flowchart LR A[Chapter 1
Random Variables] B[Chapter 2
Central Limit Theorem] C[Chapter 3
Markov Processes] D[Chapter 4
Stochastic Differential Equations] E[Chapter 5
Process Control] A --> B --> C --> D --> E style A fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style B fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style C fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style D fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff style E fill:#667eea,stroke:#764ba2,stroke-width:2px,color:#fff

📋 Learning Objectives

Understand the theory of probability variables and probability distributions and implement them in Python
Theoretically understand the law of large numbers and central limit theorem and verify them through simulation
Understand the properties of Markov processes and Poisson processes and apply them
Understand the basics of Wiener processes and stochastic differential equations and implement numerical solutions
Practice probability modeling and quality control in materials process engineering

📖 Prerequisites

Basic knowledge of calculus (integration, basic differential equations) is sufficient. Understanding of basic Python usage is desirable. Knowledge of linear algebra (matrix operations) will enable deeper understanding.

Chapter 1

Fundamentals of Random Variables and Probability Distributions

Learn about discrete and continuous random variables, probability mass functions (PMF) and probability density functions (PDF), expectation, variance, and moments, and representative distributions (binomial distribution, Poisson distribution, normal distribution, exponential distribution).

Discrete/Continuous Random Variables PMF/PDF Expectation/Variance Binomial/Poisson Distribution Normal/Exponential Distribution