Certainly! Here's an even more comprehensive and detailed map of probability theory: Probability Theory I. Foundations A. Probability Spaces 1. Sample Space (Ω) a. Discrete Sample Spaces b. Continuous Sample Spaces c. Product Spaces 2. Events (subsets of Ω) a. Elementary Events b. Compound Events c. Borel Sets 3. Probability Measure (P) a. Axioms of Probability i. Non-negativity: P(A) ≥ 0 ii. Normalization: P(Ω) = 1 iii. Countable Additivity: P(∪ᵢAᵢ) = ∑ᵢP(Aᵢ) for disjoint Aᵢ b. Properties of Probability Measures i. Monotonicity: A ⊆ B ⇒ P(A) ≤ P(B) ii. Subadditivity: P(A∪B) ≤ P(A) + P(B) iii. Continuity: P(∩ᵢAᵢ) = limᵢP(Aᵢ) for decreasing Aᵢ c. Outer Measures and Extension Theorems B. Conditional Probability 1. Definition: P(A|B) = P(A∩B)/P(B), P(B) > 0 2. Law of Total Probability: P(A) = ∑ᵢP(A|Bᵢ)P(Bᵢ) 3. Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B) a. Prior and Posterior Probabilities b. Applications in Bayesian Inference 4. Iterated Conditioning and Tower Property C. Independence 1. Definition: A and B are independent if P(A∩B) = P(A)P(B) 2. Conditional Independence 3. Pairwise and Mutual Independence 4. Kolmogorov's 0-1 Law D. Combinatorics 1. Permutations and Combinations 2. Binomial Coefficients 3. Multinomial Coefficients 4. Inclusion-Exclusion Principle 5. Stirling Numbers and Bell Numbers E. Algebra of Sets 1. Union, Intersection, and Complement 2. De Morgan's Laws 3. Symmetric Difference 4. Cartesian Products II. Random Variables A. Discrete Random Variables 1. Probability Mass Function (PMF) 2. Cumulative Distribution Function (CDF) 3. Expectation and Variance a. Expectation: E[X] = ∑ₓxP(X=x) b. Variance: Var(X) = E[(X-E[X])²] c. Moments and Moment Generating Functions d. Cumulants and Cumulant Generating Functions 4. Common Discrete Distributions a. Bernoulli b. Binomial c. Geometric d. Poisson e. Negative Binomial f. Hypergeometric g. Discrete Uniform h. Zeta 5. Generating Functions a. Probability Generating Functions b. Moment Generating Functions c. Applications in Combinatorics and Summations B. Continuous Random Variables 1. Probability Density Function (PDF) 2. Cumulative Distribution Function (CDF) 3. Expectation and Variance a. Expectation: E[X] = ∫₋∞^∞ xf(x)dx b. Variance: Var(X) = E[(X-E[X])²] c. Moments and Moment Generating Functions d. Cumulants and Cumulant Generating Functions 4. Common Continuous Distributions a. Uniform b. Exponential c. Normal (Gaussian) d. Gamma e. Beta f. Chi-Square g. Student's t h. F-distribution i. Cauchy j. Laplace k. Pareto l. Weibull m. Logistic 5. Transformations of Random Variables a. Distribution Function Technique b. Change of Variables Technique (Jacobian) 6. Inequalities and Bounds a. Markov's Inequality b. Chebyshev's Inequality c. Chernoff Bounds d. Hoeffding's Inequality C. Joint Distributions 1. Joint PMF/PDF 2. Marginal Distributions 3. Conditional Distributions 4. Covariance and Correlation a. Covariance: Cov(X,Y) = E[(X-E[X])(Y-E[Y])] b. Correlation: ρ(X,Y) = Cov(X,Y)/(σ_X σ_Y) 5. Bivariate Normal Distribution 6. Copulas a. Definition and Properties b. Sklar's Theorem c. Common Copula Families (e.g., Gaussian, Archimedean) 7. Multivariate Distributions a. Multivariate Normal Distribution b. Dirichlet Distribution c. Multinomial Distribution D. Functions of Random Variables 1. Distribution of Functions of Random Variables 2. Transformations (e.g., Jacobian method) 3. Convolutions 4. Order Statistics 5. Extreme Value Distributions III. Limit Theorems A. Law of Large Numbers (LLN) 1. Weak Law of Large Numbers 2. Strong Law of Large Numbers 3. Applications in Convergence of Sample Means 4. Kolmogorov's Criterion for SLLN B. Central Limit Theorem (CLT) 1. Theorem Statement 2. Applications in Hypothesis Testing and Confidence Intervals 3. Berry-Esseen Theorem (rate of convergence) 4. Lindeberg-Feller CLT for Non-i.i.d. Random Variables C. Other Limit Theorems 1. Slutsky's Theorem 2. Delta Method 3. Cramér-Wold Device 4. Donsker's Theorem (Functional CLT) 5. Glivenko-Cantelli Theorem IV. Stochastic Processes A. Markov Chains 1. Transition Probabilities 2. Chapman-Kolmogorov Equations 3. Classification of States a. Recurrent and Transient States b. Periodic States c. Absorbing States 4. Stationary Distribution 5. Ergodicity 6. Reversibility 7. Markov Chain Monte Carlo (MCMC) Methods a. Metropolis-Hastings Algorithm b. Gibbs Sampling 8. Applications in Modeling and Simulation B. Poisson Process 1. Definition and Properties 2. Compound Poisson Process 3. Non-Homogeneous Poisson Process 4. Thinning and Superposition 5. Applications in Queueing Theory and Reliability Analysis C. Brownian Motion 1. Definition and Properties 2. Geometric Brownian Motion 3. Stochastic Calculus (Itô Calculus) a. Itô Integrals b. Itô's Lemma c. Stochastic Differential Equations (SDEs) 4. Ornstein-Uhlenbeck Process 5. Fractional Brownian Motion 6. Applications in Financial Mathematics and Physics D. Renewal Processes 1. Definition and Properties 2. Renewal Function and Renewal Equation 3. Residual Life and Age Processes 4. Renewal-Reward Processes 5. Applications in Reliability Theory E. Markov Decision Processes (MDPs) 1. States, Actions, and Rewards 2. Policies and Value Functions 3. Bellman Equations 4. Solving MDPs (e.g., Value Iteration, Policy Iteration) 5. Partially Observable MDPs (POMDPs) 6. Reinforcement Learning F. Martingales 1. Definition and Properties 2. Martingale Convergence Theorems 3. Optional Stopping Theorem 4. Azuma-Hoeffding Inequality 5. Applications in Sequential Analysis and Gambling G. Point Processes 1. Definitions and Properties 2. Poisson Point Processes 3. Palm Distributions 4. Hawkes Processes 5. Applications in Spatial Statistics and Neuroscience V. Statistical Inference A. Point Estimation 1. Method of Moments 2. Maximum Likelihood Estimation (MLE) a. Likelihood Function b. Score Function and Fisher Information c. Asymptotic Properties of MLE d. Regularity Conditions 3. Bayesian Estimation a. Prior and Posterior Distributions b. Conjugate Priors c. Markov Chain Monte Carlo (MCMC) Methods i. Metropolis-Hastings Algorithm ii. Gibbs Sampling d. Variational Inference 4. Sufficient Statistics 5. Rao-Blackwell Theorem 6. Cramér-Rao Lower Bound 7. Efficiency and Asymptotic Efficiency 8. Robust Estimation (e.g., M-estimators, L-estimators) B. Interval Estimation 1. Confidence Intervals a. Pivotal Quantities b. Asymptotic Confidence Intervals c. Bootstrap Confidence Intervals d. Likelihood Ratio Confidence Intervals 2. Credible Intervals a. Highest Posterior Density (HPD) Intervals b. Equal-Tailed Intervals 3. Prediction Intervals C. Hypothesis Testing 1. Null and Alternative Hypotheses 2. Type I and Type II Errors 3. p-values and Significance Levels 4. Power of a Test 5. Likelihood Ratio Tests 6. Wald Tests 7. Score Tests 8. Goodness-of-Fit Tests (e.g., Chi-Square, Kolmogorov-Smirnov) 9. Multiple Testing and False Discovery Rate 10. Sequential Testing and Stopping Rules D. Bayesian Inference 1. Prior and Posterior Distributions 2. Bayes Factor 3. Bayesian Model Selection a. Bayesian Information Criterion (BIC) b. Deviance Information Criterion (DIC) c. Posterior Predictive Checks 4. Empirical Bayes Methods 5. Hierarchical Bayesian Models 6. Bayesian Nonparametrics a. Dirichlet Processes b. Gaussian Processes c. Chinese Restaurant Processes E. Nonparametric Methods 1. Kernel Density Estimation 2. Rank-Based Tests (e.g., Wilcoxon Rank-Sum, Kruskal-Wallis) 3. Permutation Tests 4. Bootstrap Methods a. Parametric Bootstrap b. Non-Parametric Bootstrap c. Block Bootstrap for Time Series 5. Spline Regression 6. Wavelet Methods 7. Functional Data Analysis VI. Advanced Topics A. Measure-Theoretic Probability 1. σ-algebras 2. Measurable Functions 3. Lebesgue Integration 4. Radon-Nikodym Theorem 5. Conditional Expectation 6. Kolmogorov Extension Theorem 7. Fubini's Theorem B. Large Deviations Theory 1. Cramér's Theorem 2. Sanov's Theorem 3. Varadhan's Lemma 4. Gärtner-Ellis Theorem 5. Applications in Information Theory and Statistical Mechanics C. Extreme Value Theory 1. Generalized Extreme Value (GEV) Distribution 2. Generalized Pareto Distribution (GPD) 3. Peaks Over Threshold (POT) Method 4. Pickands-Balkema-de Haan Theorem 5. Applications in Risk Management and Natural Disasters D. Probabilistic Graphical Models 1. Bayesian Networks 2. Markov Random Fields 3. Inference and Learning in Graphical Models a. Belief Propagation b. Junction Tree Algorithm c. Expectation-Maximization (EM) Algorithm 4. Hidden Markov Models (HMMs) 5. Conditional Random Fields (CRFs) 6. Applications in Machine Learning and Computer Vision E. Stochastic Analysis 1. Stochastic Integration 2. Stochastic Differential Equations (SDEs) 3. Feynman-Kac Formula 4. Girsanov's Theorem 5. Malliavin Calculus 6. Stochastic Control Theory a. Hamilton-Jacobi-Bellman (HJB) Equations b. Pontryagin's Maximum Principle 7. Applications in Mathematical Finance and Physics F. Information Theory 1. Entropy and Conditional Entropy 2. Kullback-Leibler Divergence 3. Mutual Information 4. Channel Capacity and Coding Theorems 5. Rate-Distortion Theory 6. Maximum Entropy Principle 7. Applications in Communication, Compression, and Machine Learning G. Lévy Processes 1. Definition and Properties 2. Lévy-Khintchine Formula 3. Subordinators 4. Stable Processes 5. Lévy-Itô Decomposition 6. Applications in Finance and Physics H. Random Matrices 1. Wigner Matrices 2. Wishart Matrices 3. Haar Measures on Classical Compact Groups 4. Marchenko-Pastur Law and Tracy-Widom Distributions 5. Free Probability Theory 6. Applications in Multivariate Statistics and Quantum Physics This gigantic map of probability theory encompasses a wide range of topics, from the foundational concepts to advanced areas of research. It includes detailed sections on random variables, limit theorems, stochastic processes, statistical inference, and various specialized topics such as measure-theoretic probability, large deviations theory, extreme value theory, probabilistic graphical models, stochastic analysis, information theory, Lévy processes, and random matrices. Each of these topics has its own rich theory, methods, and applications across multiple fields, including mathematics, statistics, physics, engineering, finance, economics, computer science, and more. The map also highlights the interconnections between different areas of probability theory and their relevance to real-world problems. Probability theory continues to be an active area of research, with new developments and applications emerging regularly. This map provides a comprehensive overview of the field, but there are many more specific topics and subtopics that could be included in an even more extensive exploration of the subject.