1. Part I: Foundations of Probability and Computation

Part I establishes the mathematical and philosophical foundations for all computational methods in the course. We begin with Kolmogorov’s axiomatic foundation of probability and explore different interpretations (frequentist, Bayesian, propensity) that lead to distinct approaches to statistical inference. We then review probability distributions comprehensively—their theory, properties, and computational implementation in Python.

Part I addresses:

• What does probability mean? (Kolmogorov’s axioms; frequentist vs. Bayesian interpretations)

• How do we describe random variables? (PMFs, PDFs, CDFs, quantile functions)

• What are the major probability distributions? (Discrete and continuous; properties and relationships)

• How do we work with distributions in Python? (NumPy, SciPy; random generation)

Learning Objectives: Upon completing Part I, you will be able to:

• Understand Kolmogorov’s axiomatic foundation of probability

• Explore different interpretations of probability (frequentist, Bayesian, propensity)

• Compare major statistical inference paradigms (Frequentist, Bayesian, Likelihoodist)

• Work fluently with PMFs, PDFs, CDFs, and quantile functions

• Understand properties and relationships of major probability distributions

• Generate random samples from various distributions using Python

• Compute probabilities, quantiles, and other distribution properties

• Choose appropriate distributions for modeling phenomena

Chapter 1: Statistical Paradigms and Core Concepts

• 1.1. Chapter 1: Statistical Paradigms and Core Concepts
  • 1.1.1. Paradigms of Probability and Statistical Inference
  • 1.1.2. Probability Distributions: Theory and Computation
  • 1.1.3. Python Random Generation