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2.8. Mathematical and Statistical Foundations (MSF)
The mathematical and statistical foundations of computer science provide the theoretical underpinnings for the discipline. They include discrete mathematics, probability, statistics, linear algebra, and calculus. These areas are essential for understanding algorithms, data structures, machine learning, graphics, cryptography, and many other topics in computer science.
| Knowledge Area (KA) | CS Core | KA Core |
 2.8.1 Discrete Mathematics | 15 | 6 |
|
2.8.2 Advanced Discrete Mathematics | 15 | 6 |
|
2.8.3 Probability | 11 | 29 |
|
2.8.4 Statistics | 10 | 30 |
|
2.8.5 Linear Algebra | 5 | 35 |
|
2.8.6 Calculus | 40 |
2.8.1. MSF/Discrete Mathematics (CS Core: 15 hrs, KA Core: 6 hrs) ↑ Back to top
Topics:
Core
- Sets: operations, cardinality, power sets, Cartesian products
- Relations: properties, equivalence relations, partial orders
- Functions: injective, surjective, bijective, composition, inverse
- Logic: propositional and predicate logic, quantifiers, proofs
- Graphs: definitions, paths, cycles, trees, connectivity
- Counting: permutations, combinations, binomial theorem
- Recurrence relations: solving simple recurrences
Learning Outcomes:
Core:
- Define basic set operations and relations [Explain]
- Apply set operations to solve problems [Apply]
- Define properties of relations [Explain]
- Determine if a relation is an equivalence relation [Apply]
- Define types of functions [Explain]
- Compose functions and find inverses [Apply]
- Explain propositional and predicate logic [Explain]
- Construct simple logical proofs [Apply]
- Define basic graph concepts [Explain]
- Solve basic graph problems [Apply]
- Explain permutations and combinations [Explain]
- Apply counting techniques to solve problems [Apply]
- Define recurrence relations [Explain]
- Solve simple recurrence relations [Apply]
2.8.2. MSF/Advanced Discrete Mathematics (CS Core: 15 hrs, KA Core: 6 hrs) ↑ Back to top
Topics:
Core
- Advanced set theory: infinite sets, Cantor's diagonal argument
- Advanced relations: closures, lattices
- Advanced functions: recursive functions, computability
- Advanced logic: proof systems, completeness, decidability
- Advanced graph theory: planarity, coloring, networks
- Advanced counting: generating functions, inclusion-exclusion
- Advanced recurrence: generating functions, asymptotic behavior
Learning Outcomes:
Core:
- Explain concepts of infinite sets and cardinality [Explain]
- Apply closure operations to relations [Apply]
- Analyze advanced graph properties [Evaluate]
- Develop generating functions for counting problems [Create]
2.8.3. MSF/Probability (CS Core: 11 hrs, KA Core: 29 hrs) ↑ Back to top
Topics:
Core
- Basic probability: sample spaces, events, axioms
- Conditional probability, independence, Bayes' theorem
- Random variables: discrete and continuous, distributions
- Expectation, variance, standard deviation
- Common distributions: uniform, binomial, normal
- Joint distributions, covariance, correlation
- Limit theorems: law of large numbers, central limit theorem
- Stochastic processes: Markov chains
- Information theory: entropy, mutual information
Learning Outcomes:
Core:
- Define basic probability concepts [Explain]
- Calculate probabilities of events [Apply]
- Define conditional probability and independence [Explain]
- Apply Bayes' theorem to update probabilities [Apply]
- Define random variables and distributions [Explain]
- Compute expectation and variance [Apply]
- Describe common probability distributions [Explain]
- Apply common distributions to model real-world phenomena [Apply]
- Explain joint distributions and dependence [Explain]
- Compute covariance and correlation [Apply]
- Analyze the implications of limit theorems [Evaluate]
- Model simple systems using Markov chains [Create]
2.8.4. MSF/Statistics (CS Core: 10 hrs, KA Core: 30 hrs) ↑ Back to top
Topics:
Core
- Descriptive statistics: measures of center and spread
- Data visualization: histograms, box plots, scatter plots
- Sampling distributions: sampling variability, central limit theorem
- Estimation: point estimates, confidence intervals
- Hypothesis testing: null and alternative hypotheses, p-values
- Regression: simple and multiple linear regression
- Analysis of variance (ANOVA)
- Nonparametric methods
- Experimental design: randomization, control, blinding
Learning Outcomes:
Core:
- Define descriptive statistics measures [Explain]
- Compute descriptive statistics for a dataset [Apply]
- Explain common data visualization techniques [Explain]
- Create appropriate visualizations for given data [Apply]
- Define sampling distributions and variability [Explain]
- Apply the central limit theorem to approximate distributions [Apply]
- Define point estimates and confidence intervals [Explain]
- Compute confidence intervals for means and proportions [Apply]
- Define hypothesis testing concepts [Explain]
- Conduct a simple hypothesis test and interpret p-values [Apply]
- Explain simple and multiple linear regression [Explain]
- Fit a regression model and interpret coefficients [Apply]
- Analyze variance using ANOVA [Evaluate]
- Design a simple experiment with appropriate controls [Create]
2.8.5. MSF/Linear Algebra (CS Core: 5 hrs, KA Core: 35 hrs) ↑ Back to top
Topics:
Core
- Vectors: operations, dot product, cross product
- Matrices: operations, inverses, determinants
- Systems of linear equations: Gaussian elimination, matrix form
- Vector spaces: subspaces, basis, dimension
- Linear transformations: matrix representation, kernel, image
- Eigenvalues and eigenvectors: diagonalization
- Orthogonality: orthonormal bases, Gram-Schmidt
- Singular value decomposition (SVD)
- Applications: least squares, computer graphics, machine learning
Learning Outcomes:
Core:
- Define vector operations and properties [Explain]
- Compute vector operations and products [Apply]
- Define matrix operations and properties [Explain]
- Compute matrix operations, inverses, and determinants [Apply]
- Explain systems of linear equations and solution methods [Explain]
- Solve systems of linear equations using elimination [Apply]
- Define vector spaces, subspaces, basis, and dimension [Explain]
- Determine if a set of vectors forms a basis [Apply]
- Define linear transformations and their representations [Explain]
- Find the kernel and image of a linear transformation [Apply]
- Explain eigenvalues and eigenvectors [Explain]
- Diagonalize a matrix when possible [Apply]
- Analyze orthogonality and orthonormal bases [Evaluate]
- Apply singular value decomposition to a matrix [Create]
2.8.6. MSF/Calculus (KA Core: 40 hrs) ↑ Back to top
Topics:
Core
- Limits and continuity
- Derivatives: definition, rules, applications
- Integrals: definite and indefinite, fundamental theorem
- Series: convergence, Taylor series
- Multivariable calculus: partial derivatives, gradients
- Optimization: local and global extrema, Lagrange multipliers
- Differential equations: basic solution techniques
- Numerical methods: integration, differentiation
Learning Outcomes:
Core:
- Define limits and continuity [Explain]
- Compute limits of functions [Apply]
- Define derivatives and differentiation rules [Explain]
- Compute derivatives of functions [Apply]
- Define integrals and the fundamental theorem of calculus [Explain]
- Compute definite and indefinite integrals [Apply]
- Define series and convergence [Explain]
- Compute Taylor series expansions [Apply]
- Explain partial derivatives and gradients [Explain]
- Compute partial derivatives and gradients [Apply]
- Analyze optimization problems using calculus [Evaluate]
- Solve simple differential equations [Create]