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3.4. Probability and Statistics (PST)

This area covers the mathematical theory of uncertainty and inference, from probability axioms and stochastic processes to Bayesian inference and statistical learning, which are essential foundations for AI, data science, and machine learning.

Knowledge Area (KA)	CS Core	KA Core
3.4.1 Probability Axioms and Distributions	Elective
3.4.2 Limit Theorems	Elective
3.4.3 Stochastic Processes	Elective
3.4.4 Estimation Theory	Elective
3.4.5 Bayesian Inference	Elective

3.4.1. PST/Probability Axioms and Distributions ↑ Back to top

Kolmogorov's axiomatic foundation of probability, random variables, distributions, and expectations, with applications in algorithm analysis and AI.
Topics:
Core

• Kolmogorov axioms, sigma-algebras, and probability spaces
• Random variables: discrete and continuous distributions, CDF, PDF, and PMF
• Expectation, variance, moments, and moment generating functions
• Joint, marginal, and conditional distributions; independence and covariance
• Standard families: Binomial, Poisson, Normal, Exponential, Gamma, and Beta

Learning Outcomes:
Core:

1. State Kolmogorov's axioms and construct a probability space for a standard experiment [Familiarity]
2. Compute expectations, variances, and moment generating functions for standard distributions [Usage]
3. Determine joint and conditional distributions and assess independence of random variables [Assessment]

3.4.2. PST/Limit Theorems ↑ Back to top

Laws of large numbers, central limit theorem, modes of convergence, and large deviations, foundational for algorithm analysis and statistical inference.
Topics:
Core

• Modes of convergence: almost sure, in probability, in \(L^p\), and in distribution
• Weak and strong laws of large numbers and their applications
• Central Limit Theorem: statement, proof via characteristic functions, and Berry-Esseen bound
• Characteristic functions and the continuity theorem (Lévy)

Learning Outcomes:
Core:

1. Distinguish between modes of convergence of random variables and provide counterexamples [Familiarity]
2. Apply the Central Limit Theorem to approximate distributions of sums and means [Usage]
3. Prove the strong law of large numbers and estimate large deviation probabilities [Assessment]

3.4.3. PST/Stochastic Processes ↑ Back to top

Markov chains, Poisson processes, Brownian motion, martingales, and stochastic calculus, with applications to queuing, simulation, and reinforcement learning.
Topics:
Core

• Discrete-time Markov chains: classification of states, stationary distributions, and ergodic theorem
• Poisson processes: definition, properties, and generalizations (compound, inhomogeneous)
• Brownian motion: definition, path properties, and the reflection principle
• Martingales, stopping times, optional sampling theorem, and Doob's inequalities

Learning Outcomes:
Core:

1. Classify states of a Markov chain and determine its stationary distribution [Familiarity]
2. Apply martingale theory and the optional sampling theorem to solve stopping time problems [Usage]
3. Model a queueing system using Poisson arrival processes and analyze its steady-state distribution [Assessment]

3.4.4. PST/Estimation Theory ↑ Back to top

Point and interval estimation: sufficiency, maximum likelihood, and asymptotic theory, used throughout statistical machine learning and data science.
Topics:
Core

• Sufficient statistics, the factorization theorem, and completeness
• Unbiased estimators, the Cramér-Rao lower bound, and UMVUE
• Maximum likelihood estimation: properties, consistency, and asymptotic normality
• Confidence intervals: pivotal quantities, likelihood ratio, and bootstrap methods
• Asymptotic theory: delta method, M-estimators, and the Fisher information matrix

Learning Outcomes:
Core:

1. Explain the Cramér-Rao bound and identify when a UMVUE exists [Familiarity]
2. Derive maximum likelihood estimators and compute their asymptotic variance [Usage]
3. Construct confidence intervals using pivotal quantities and bootstrap resampling [Assessment]

3.4.5. PST/Bayesian Inference ↑ Back to top

Bayesian paradigm: prior and posterior distributions, conjugate families, hierarchical models, and MCMC methods, central to probabilistic machine learning and AI.
Topics:
Core

• Bayes' theorem in continuous settings; prior and posterior distributions
• Conjugate prior families: Beta-Binomial, Dirichlet-Multinomial, Normal-Normal
• Hierarchical Bayesian models and empirical Bayes
• Markov chain Monte Carlo (MCMC): Metropolis-Hastings and Gibbs sampling
• Variational inference and the evidence lower bound (ELBO)

Learning Outcomes:
Core:

1. Apply Bayes' theorem to update a prior with observed data and obtain a posterior distribution [Familiarity]
2. Construct a hierarchical Bayesian model and derive its joint and conditional distributions [Usage]
3. Implement Gibbs sampling or Metropolis-Hastings to approximate a posterior distribution [Assessment]