Lesson Directory
Discrete and Stochastic Relationships develops reasoning about countable objects, finite structures, algorithms, randomness, probability, data, and inference. Lessons are grouped below by section.
Sections
4.1 Discrete Objects and Structures
- Finite and Countably Infinite Sets
- Tuples, Strings, Lists, and Sequences
- Arrangements, Selections, and Partitions
- Relations and Finite Mappings
- Graphs, Trees, Networks, and States
- Tables, Arrays, and Incidence Structures
- Discrete Functions and Stepwise Change
- Configurations and State Spaces
- Equivalence Classes
- Structural Invariants
4.2 Discrete Reasoning and Proof
4.3 Fundamental Counting Principles
4.4 Permutations, Combinations, and Arrangements
4.5 Sequences, Recurrences, and Generating Structures
- Explicit and Recursive Sequences
- Initial Conditions
- Arithmetic and Geometric Recurrences
- Linear Recurrence Relations
- Fibonacci-Type Sequences
- Difference Equations
- Iteration and State Transitions
- Closed Forms
- Introductory Characteristic Methods
- Generating Functions
- Recursion Trees
- Combinatorial Interpretations of Sequences
4.6 Number Theory and Modular Structures
- Divisibility
- Prime and Composite Numbers
- Greatest Common Divisors and Least Common Multiples
- The Euclidean Algorithm
- Prime Factorization
- Congruence and Modular Arithmetic
- Residue Classes
- Linear Congruences
- Diophantine Equations
- Fermat and Euler Relationships
- Introductory Cryptography
- Integer Patterns and Arithmetic Functions
4.7 Graphs, Trees, and Networks
- Vertices, Edges, Adjacency, and Incidence
- Directed and Undirected Graphs
- Weighted Graphs
- Paths, Walks, Trails, Circuits, and Connectivity
- Trees and Rooted Trees
- Spanning Trees
- Eulerian and Hamiltonian Problems
- Planarity and Graph Coloring
- Shortest Paths
- Network Flows
- Bipartite Graphs and Matching
- Social, Transportation, Communication, and Dependency Networks
4.8 Discrete Algorithms and Computation
4.9 Logic, Boolean Structures, and Finite Systems
4.10 Probability Foundations
- Experiments, Outcomes, and Sample Spaces
- Events and Event Operations
- Axioms of Probability
- Equally Likely Outcomes
- Complement, Addition, and Multiplication Rules
- Counting-Based Probability
- Geometric Probability
- Conditional Probability
- Independence
- Simulation and Empirical Probability
- Long-Run Frequency
- Model Assumptions and Limitations
4.11 Random Variables and Probability Distributions
- Discrete and Continuous Random Variables
- Probability Mass and Density Functions
- Cumulative Distributions
- Expected Value
- Variance and Standard Deviation
- Bernoulli and Binomial Distributions
- Geometric and Negative Binomial Distributions
- Hypergeometric and Poisson Distributions
- Uniform, Normal, and Exponential Distributions
- Joint Distributions
- Transformations of Random Variables
- Law of Large Numbers and Central Limit Theorem
4.12 Dependence, Conditioning, and Bayesian Reasoning
4.13 Stochastic Processes
4.14 Data and Descriptive Statistics
- Data Types and Measurement Levels
- Populations, Samples, and Variables
- Frequency Tables
- Categorical and Quantitative Displays
- Mean, Median, Mode, and Quantiles
- Range, Variance, Standard Deviation, and Interquartile Range
- Shape, Center, Spread, and Outliers
- Standardization and Z-Scores
- Bivariate Data
- Covariance and Correlation
- Robust Summaries
- Misleading Displays and Data Quality
4.15 Sampling and Statistical Inference
- Random Sampling
- Sampling Distributions
- Bias, Variability, and Representativeness
- Point Estimation
- Confidence Intervals
- Hypothesis Tests
- Null and Alternative Hypotheses
- Significance Levels and P-Values
- Type I and Type II Errors
- Power and Sample Size
- Parametric and Nonparametric Methods
- Practical versus Statistical Significance
4.16 Experimental Design and Causal Reasoning
- Observational Studies and Experiments
- Treatments, Controls, and Outcomes
- Random Assignment
- Replication
- Blocking and Stratification
- Blinding
- Confounding and Lurking Variables
- Selection and Measurement Bias
- Introductory Causal Diagrams
- Counterfactual Reasoning
- Internal and External Validity
- Ethical Design and Interpretation
4.17 Statistical Relationships and Prediction
- Scatterplots and Association
- Correlation
- Simple Linear Regression
- Introductory Multiple Regression
- Residuals and Model Diagnostics
- Categorical Predictors
- Nonlinear Relationships
- Classification
- Prediction Intervals
- Overfitting and Underfitting
- Training and Validation Data
- Association versus Causation
4.18 Decision, Risk, and Information
- Decision Tables and Decision Trees
- Expected Value and Expected Utility
- Loss Functions
- Risk and Uncertainty
- Sensitivity Analysis
- Value of Information
- Entropy and Information
- Coding and Compression
- Reliability and Failure Probability
- Games and Strategic Decisions
- Portfolio and Allocation
- Ethical Communication of Risk
4.19 Applied Discrete and Stochastic Problem Families
- Scheduling, Assignment, and Routing
- Voting and Apportionment
- Networks and Communication
- Coding, Security, and Cryptography
- Inventory and Queueing
- Reliability and Quality Control
- Games and Strategic Behavior
- Biological and Population Processes
- Finance, Insurance, and Risk
- Medical Testing and Public-Health Inference
- Surveys, Experiments, and Forecasting
- Simulation and Data-Driven Decision-Making