0.1An Interactive First Course¶
Dr. Anjana Yatawara
1How This Book Works¶
Every concept in this book follows a three-step pipeline:
See it. We start with a simulation or a concrete example so you can observe the phenomenon before we formalize it.
Prove it. We prove every result rigorously. No “it can be shown that.” No hand-waving. The proof is where the learning happens.
Simulate it. After the proof, R code confirms the result empirically. You’ll see theory and practice agree — and when they don’t, you’ll understand why.
2What You’ll Learn¶
By the end of this book, you will be able to:
Construct probability models for random experiments and compute event probabilities using axioms and counting methods
Derive and apply the major discrete and continuous probability distributions
Work with joint, marginal, and conditional distributions for multiple random variables
Compute expected values, variances, covariances, and moment generating functions
Find the distribution of functions and transformations of random variables
Write R code to simulate, visualize, and verify every result you prove by hand
3The Chapters at a Glance¶
| Chapter | Title | The Big Question |
|---|---|---|
| 1 | The Big Picture | What is probability theory, and why should I care? |
| 2 | Probability Foundations | How do we measure the likelihood of events? |
| 3 | Conditional Probability & Bayes | How does new information change what we believe? |
| 4 | Discrete Random Variables | How do we model outcomes we can count? |
| 5 | Continuous Random Variables | How do we model outcomes on a continuum? |
| 6 | Multivariate Distributions | What happens when multiple random variables interact? |
| 7 | Functions of Random Variables | If we transform a random variable, what’s the new distribution? |
4About This Book¶
This textbook is designed as the primary resource for an undergraduate probability theory course (typically titled “Probability Theory” or “Mathematical Statistics I”). It closely follows the first six chapters of the standard curriculum while adding:
A Chapter 1 crash course in basic statistics for students without an intro stats prerequisite
Real-world hooks that open every chapter with a famous problem or dramatic application
R simulations alongside every major result
50-question quiz games at the end of every chapter for self-assessment
Guided practice problems with hidden solutions throughout each chapter
5How to Cite This Book¶
Yatawara, A. (2026). Probability from the Ground Up: An Interactive First Course. [Online textbook]. https://
yatawara .com /math3200
6License¶
This work is licensed under Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). You are free to share and adapt this material for non-commercial purposes, provided you give appropriate attribution.
Content current as of April 2026.