Chapter 1: Statistical Paradigms and Core Concepts

This chapter establishes the foundation for computational data science by covering three essential topics:

  1. Probability foundations and inference paradigms (Section 1.1): Kolmogorov’s axioms and philosophical interpretations of probability; comparison of frequentist, Bayesian, and likelihood-based inference approaches

  2. Probability distributions (Section 1.2): Comprehensive review of discrete and continuous distributions, their properties, relationships, and applications; detailed treatment of PMFs, PDFs, and CDFs

  3. Python random generation (Section 1.3): Practical implementation of probability concepts using Python’s ecosystem (random, NumPy, SciPy)

Learning Objectives: Upon completion of this chapter, students will be able to:

  • Understand Kolmogorov’s axiomatic foundation of probability

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

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

  • Recognize philosophical debates and practical trade-offs between approaches

  • Work fluently with random variables, PMFs, PDFs, and CDFs

  • Understand key properties and applications of major probability distributions

  • Generate random samples from various distributions using Python

  • Compute probabilities, quantiles, and other distribution properties

  • Choose appropriate Python library and distribution for specific problems

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