Data Science

  1. Mathematical and Statistical Foundations
    1. Linear Algebra
      1. Vectors and Vector Operations
        1. Vector Definition
          1. Vector Addition and Subtraction
            1. Scalar Multiplication
              1. Dot Product
                1. Cross Product
                  1. Vector Norms
                  2. Vector Spaces
                    1. Vector Space Definition
                      1. Subspaces
                        1. Linear Independence
                          1. Basis and Dimension
                            1. Span
                            2. Matrices and Matrix Operations
                              1. Matrix Definition
                                1. Matrix Addition and Subtraction
                                  1. Matrix Multiplication
                                    1. Transpose
                                      1. Inverse
                                        1. Determinants
                                          1. Rank
                                          2. Systems of Linear Equations
                                            1. Gaussian Elimination
                                              1. Matrix Form of Linear Systems
                                                1. Solution Methods
                                                2. Eigenvalues and Eigenvectors
                                                  1. Definitions
                                                    1. Characteristic Equation
                                                      1. Eigendecomposition
                                                        1. Diagonalization
                                                          1. Applications in Data Science
                                                          2. Singular Value Decomposition
                                                            1. SVD Definition
                                                              1. Computing SVD
                                                                1. Applications in Dimensionality Reduction
                                                                  1. Applications in Recommendation Systems
                                                                2. Calculus
                                                                  1. Limits and Continuity
                                                                    1. Limit Definition
                                                                      1. Continuity
                                                                        1. Applications to Optimization
                                                                        2. Derivatives
                                                                          1. Derivative Definition
                                                                            1. Differentiation Rules
                                                                              1. Chain Rule
                                                                                1. Applications in Machine Learning
                                                                                2. Partial Derivatives
                                                                                  1. Multivariable Functions
                                                                                    1. Partial Derivative Definition
                                                                                      1. Gradient Vectors
                                                                                        1. Directional Derivatives
                                                                                        2. Optimization
                                                                                          1. Critical Points
                                                                                            1. Local and Global Extrema
                                                                                              1. Lagrange Multipliers
                                                                                                1. Gradient Descent
                                                                                                  1. Stochastic Gradient Descent
                                                                                                    1. Learning Rate and Convergence
                                                                                                    2. Multivariable Calculus
                                                                                                      1. Jacobian Matrices
                                                                                                        1. Hessian Matrices
                                                                                                          1. Taylor Series Expansion
                                                                                                        2. Probability Theory
                                                                                                          1. Basic Probability Concepts
                                                                                                            1. Sample Spaces and Events
                                                                                                              1. Probability Axioms
                                                                                                                1. Probability Rules
                                                                                                                  1. Combinatorics
                                                                                                                  2. Conditional Probability
                                                                                                                    1. Independence
                                                                                                                      1. Multiplication Rule
                                                                                                                        1. Law of Total Probability
                                                                                                                        2. Bayes' Theorem
                                                                                                                          1. Statement and Proof
                                                                                                                            1. Applications in Machine Learning
                                                                                                                              1. Naive Bayes Classifier Foundation
                                                                                                                              2. Random Variables
                                                                                                                                1. Discrete Random Variables
                                                                                                                                  1. Continuous Random Variables
                                                                                                                                    1. Probability Mass Functions
                                                                                                                                      1. Probability Density Functions
                                                                                                                                        1. Cumulative Distribution Functions
                                                                                                                                        2. Expected Value and Variance
                                                                                                                                          1. Expected Value Definition
                                                                                                                                            1. Properties of Expected Value
                                                                                                                                              1. Variance Definition
                                                                                                                                                1. Properties of Variance
                                                                                                                                                  1. Covariance
                                                                                                                                                    1. Correlation
                                                                                                                                                    2. Common Probability Distributions
                                                                                                                                                      1. Discrete Distributions
                                                                                                                                                        1. Bernoulli Distribution
                                                                                                                                                          1. Binomial Distribution
                                                                                                                                                            1. Poisson Distribution
                                                                                                                                                              1. Geometric Distribution
                                                                                                                                                                1. Negative Binomial Distribution
                                                                                                                                                                2. Continuous Distributions
                                                                                                                                                                  1. Uniform Distribution
                                                                                                                                                                    1. Normal Distribution
                                                                                                                                                                      1. Exponential Distribution
                                                                                                                                                                        1. Gamma Distribution
                                                                                                                                                                          1. Beta Distribution
                                                                                                                                                                            1. Chi-Square Distribution
                                                                                                                                                                              1. t-Distribution
                                                                                                                                                                                1. F-Distribution
                                                                                                                                                                              2. Joint Distributions
                                                                                                                                                                                1. Joint Probability Mass Functions
                                                                                                                                                                                  1. Joint Probability Density Functions
                                                                                                                                                                                    1. Marginal Distributions
                                                                                                                                                                                      1. Independence of Random Variables
                                                                                                                                                                                      2. Central Limit Theorem
                                                                                                                                                                                        1. Statement
                                                                                                                                                                                          1. Sampling Distributions
                                                                                                                                                                                        2. Descriptive Statistics
                                                                                                                                                                                          1. Measures of Central Tendency
                                                                                                                                                                                            1. Mean
                                                                                                                                                                                              1. Median
                                                                                                                                                                                                1. Mode
                                                                                                                                                                                                  1. Geometric Mean
                                                                                                                                                                                                    1. Harmonic Mean
                                                                                                                                                                                                    2. Measures of Dispersion
                                                                                                                                                                                                      1. Range
                                                                                                                                                                                                        1. Variance
                                                                                                                                                                                                          1. Standard Deviation
                                                                                                                                                                                                            1. Interquartile Range
                                                                                                                                                                                                              1. Mean Absolute Deviation
                                                                                                                                                                                                                1. Coefficient of Variation
                                                                                                                                                                                                                2. Measures of Shape
                                                                                                                                                                                                                  1. Skewness
                                                                                                                                                                                                                    1. Kurtosis
                                                                                                                                                                                                                    2. Measures of Position
                                                                                                                                                                                                                      1. Percentiles
                                                                                                                                                                                                                        1. Quartiles
                                                                                                                                                                                                                          1. Z-scores
                                                                                                                                                                                                                          2. Correlation and Association
                                                                                                                                                                                                                            1. Pearson Correlation Coefficient
                                                                                                                                                                                                                              1. Spearman Rank Correlation
                                                                                                                                                                                                                                1. Kendall's Tau
                                                                                                                                                                                                                                  1. Covariance
                                                                                                                                                                                                                                    1. Correlation vs Causation
                                                                                                                                                                                                                                  2. Inferential Statistics
                                                                                                                                                                                                                                    1. Sampling Theory
                                                                                                                                                                                                                                      1. Population vs Sample
                                                                                                                                                                                                                                        1. Sampling Methods
                                                                                                                                                                                                                                          1. Simple Random Sampling
                                                                                                                                                                                                                                            1. Stratified Sampling
                                                                                                                                                                                                                                              1. Cluster Sampling
                                                                                                                                                                                                                                                1. Systematic Sampling
                                                                                                                                                                                                                                                2. Sampling Bias
                                                                                                                                                                                                                                                  1. Sampling Distributions
                                                                                                                                                                                                                                                  2. Estimation
                                                                                                                                                                                                                                                    1. Point Estimation
                                                                                                                                                                                                                                                      1. Properties of Estimators
                                                                                                                                                                                                                                                        1. Method of Moments
                                                                                                                                                                                                                                                          1. Maximum Likelihood Estimation
                                                                                                                                                                                                                                                            1. Interval Estimation
                                                                                                                                                                                                                                                            2. Confidence Intervals
                                                                                                                                                                                                                                                              1. Construction
                                                                                                                                                                                                                                                                1. Interpretation
                                                                                                                                                                                                                                                                  1. Confidence Intervals for Means
                                                                                                                                                                                                                                                                    1. Confidence Intervals for Proportions
                                                                                                                                                                                                                                                                      1. Confidence Intervals for Differences
                                                                                                                                                                                                                                                                      2. Hypothesis Testing
                                                                                                                                                                                                                                                                        1. Null and Alternative Hypotheses
                                                                                                                                                                                                                                                                          1. Test Statistics
                                                                                                                                                                                                                                                                            1. p-values
                                                                                                                                                                                                                                                                              1. Significance Levels
                                                                                                                                                                                                                                                                                1. Type I and Type II Errors
                                                                                                                                                                                                                                                                                  1. Statistical Power
                                                                                                                                                                                                                                                                                    1. One-tailed vs Two-tailed Tests
                                                                                                                                                                                                                                                                                    2. Common Statistical Tests
                                                                                                                                                                                                                                                                                      1. t-tests
                                                                                                                                                                                                                                                                                        1. One-sample t-test
                                                                                                                                                                                                                                                                                          1. Two-sample t-test
                                                                                                                                                                                                                                                                                            1. Paired t-test
                                                                                                                                                                                                                                                                                            2. Chi-square Tests
                                                                                                                                                                                                                                                                                              1. Goodness of Fit
                                                                                                                                                                                                                                                                                                1. Test of Independence
                                                                                                                                                                                                                                                                                                2. ANOVA
                                                                                                                                                                                                                                                                                                  1. One-way ANOVA
                                                                                                                                                                                                                                                                                                    1. Two-way ANOVA
                                                                                                                                                                                                                                                                                                    2. Non-parametric Tests
                                                                                                                                                                                                                                                                                                      1. Mann-Whitney U Test
                                                                                                                                                                                                                                                                                                        1. Wilcoxon Signed-rank Test
                                                                                                                                                                                                                                                                                                          1. Kruskal-Wallis Test
                                                                                                                                                                                                                                                                                                        2. Experimental Design
                                                                                                                                                                                                                                                                                                          1. Principles of Experimental Design
                                                                                                                                                                                                                                                                                                            1. Randomization
                                                                                                                                                                                                                                                                                                              1. Blocking
                                                                                                                                                                                                                                                                                                                1. Factorial Designs
                                                                                                                                                                                                                                                                                                                2. A/B Testing
                                                                                                                                                                                                                                                                                                                  1. Design Principles
                                                                                                                                                                                                                                                                                                                    1. Sample Size Calculation
                                                                                                                                                                                                                                                                                                                      1. Statistical Analysis
                                                                                                                                                                                                                                                                                                                        1. Interpreting Results
                                                                                                                                                                                                                                                                                                                          1. Multiple Testing Corrections

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                                                                                                                                                                                                                                                                                                                      1. Foundations of Data Science

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                                                                                                                                                                                                                                                                                                                      3. Computational Foundations and Tools