Statistics for Data Science

  1. Probability Distributions
    1. Random Variables
      1. Definition and Concept
        1. Mapping from Sample Space to Real Numbers
          1. Notation and Conventions
            1. Types of Random Variables
            2. Discrete Random Variables
              1. Definition and Characteristics
                1. Countable Outcomes
                  1. Finite or Countably Infinite Range
                  2. Probability Mass Function (PMF)
                    1. Definition and Properties
                      1. PMF Requirements
                        1. Graphical Representation
                        2. Cumulative Distribution Function (CDF)
                          1. Definition for Discrete Variables
                            1. Step Function Characteristics
                              1. Relationship to PMF
                              2. Expected Value (Mean)
                                1. Definition and Calculation
                                  1. Linearity of Expectation
                                    1. Interpretation
                                    2. Variance and Standard Deviation
                                      1. Definition and Calculation
                                        1. Alternative Formulas
                                      2. Continuous Random Variables
                                        1. Definition and Characteristics
                                          1. Uncountable Outcomes
                                            1. Continuous Range
                                            2. Probability Density Function (PDF)
                                              1. Definition and Properties
                                                1. Area Under Curve Interpretation
                                                  1. PDF Requirements
                                                  2. Cumulative Distribution Function (CDF)
                                                    1. Definition for Continuous Variables
                                                      1. Relationship to PDF
                                                        1. Properties and Characteristics
                                                        2. Expected Value (Mean)
                                                          1. Integration Formula
                                                            1. Interpretation
                                                            2. Variance and Standard Deviation
                                                              1. Integration Formulas
                                                            3. Functions of Random Variables
                                                              1. Linear Transformations
                                                                1. Expected Value of Functions
                                                                  1. Variance of Transformations
                                                                2. Discrete Probability Distributions
                                                                  1. Bernoulli Distribution
                                                                    1. Definition and Context
                                                                      1. Single Trial Success/Failure
                                                                        1. Parameter p
                                                                        2. Probability Mass Function
                                                                          1. Properties
                                                                            1. Mean (Expected Value)
                                                                              1. Variance
                                                                                1. Standard Deviation
                                                                                2. Applications
                                                                                  1. Binary Outcomes
                                                                                    1. Foundation for Other Distributions
                                                                                  2. Binomial Distribution
                                                                                    1. Definition and Context
                                                                                      1. Fixed Number of Independent Trials
                                                                                        1. Constant Success Probability
                                                                                        2. Parameters
                                                                                          1. Number of Trials (n)
                                                                                            1. Success Probability (p)
                                                                                            2. Probability Mass Function
                                                                                              1. Formula and Calculation
                                                                                                1. Combinatorial Component
                                                                                                2. Properties
                                                                                                  1. Mean (np)
                                                                                                    1. Variance (np(1-p))
                                                                                                      1. Standard Deviation
                                                                                                      2. Applications
                                                                                                        1. Quality Control
                                                                                                          1. Survey Sampling
                                                                                                            1. A/B Testing
                                                                                                            2. Normal Approximation
                                                                                                              1. Conditions for Approximation
                                                                                                                1. Continuity Correction
                                                                                                              2. Poisson Distribution
                                                                                                                1. Definition and Context
                                                                                                                  1. Rare Events in Fixed Intervals
                                                                                                                    1. Rate Parameter λ
                                                                                                                    2. Probability Mass Function
                                                                                                                      1. Formula and Calculation
                                                                                                                        1. e^(-λ) Component
                                                                                                                        2. Properties
                                                                                                                          1. Mean (λ)
                                                                                                                            1. Variance (λ)
                                                                                                                              1. Standard Deviation
                                                                                                                              2. Poisson Process
                                                                                                                                1. Assumptions and Conditions
                                                                                                                                  1. Time/Space Intervals
                                                                                                                                  2. Poisson Approximation to Binomial
                                                                                                                                    1. Conditions (large n, small p)
                                                                                                                                      1. λ = np Relationship
                                                                                                                                      2. Applications
                                                                                                                                        1. Call Center Arrivals
                                                                                                                                          1. Defect Counting
                                                                                                                                            1. Website Traffic
                                                                                                                                          2. Geometric Distribution
                                                                                                                                            1. Definition and Context
                                                                                                                                              1. First Success in Sequence
                                                                                                                                                1. Independent Trials
                                                                                                                                                2. Probability Mass Function
                                                                                                                                                  1. Formula and Calculation
                                                                                                                                                    1. (1-p)^(k-1) * p Structure
                                                                                                                                                    2. Properties
                                                                                                                                                      1. Mean (1/p)
                                                                                                                                                        1. Variance ((1-p)/p²)
                                                                                                                                                          1. Memoryless Property
                                                                                                                                                          2. Applications
                                                                                                                                                            1. Waiting Time Problems
                                                                                                                                                              1. Reliability Analysis
                                                                                                                                                            2. Negative Binomial Distribution
                                                                                                                                                              1. Definition and Context
                                                                                                                                                                1. Parameters and PMF
                                                                                                                                                                  1. Relationship to Geometric Distribution
                                                                                                                                                                  2. Hypergeometric Distribution
                                                                                                                                                                    1. Definition and Context
                                                                                                                                                                      1. Sampling Without Replacement
                                                                                                                                                                        1. Parameters and PMF
                                                                                                                                                                      2. Continuous Probability Distributions
                                                                                                                                                                        1. Uniform Distribution
                                                                                                                                                                          1. Definition and Context
                                                                                                                                                                            1. Equal Probability Over Interval
                                                                                                                                                                              1. Parameters a and b
                                                                                                                                                                              2. Probability Density Function
                                                                                                                                                                                1. Rectangular Shape
                                                                                                                                                                                  1. Height = 1/(b-a)
                                                                                                                                                                                  2. Cumulative Distribution Function
                                                                                                                                                                                    1. Linear Function
                                                                                                                                                                                      1. Calculation Methods
                                                                                                                                                                                      2. Properties
                                                                                                                                                                                        1. Mean ((a+b)/2)
                                                                                                                                                                                          1. Variance ((b-a)²/12)
                                                                                                                                                                                          2. Applications
                                                                                                                                                                                            1. Random Number Generation
                                                                                                                                                                                              1. Modeling Uncertainty
                                                                                                                                                                                            2. Normal (Gaussian) Distribution
                                                                                                                                                                                              1. Definition and Importance
                                                                                                                                                                                                1. Bell-Shaped Curve
                                                                                                                                                                                                  1. Central Limit Theorem Connection
                                                                                                                                                                                                  2. Parameters
                                                                                                                                                                                                    1. Mean (μ)
                                                                                                                                                                                                      1. Standard Deviation (σ)
                                                                                                                                                                                                      2. Probability Density Function
                                                                                                                                                                                                        1. Mathematical Formula
                                                                                                                                                                                                          1. Exponential Component
                                                                                                                                                                                                          2. Properties and Characteristics
                                                                                                                                                                                                            1. Symmetry About Mean
                                                                                                                                                                                                              1. Inflection Points
                                                                                                                                                                                                                1. Asymptotic Behavior
                                                                                                                                                                                                                2. The Standard Normal Distribution
                                                                                                                                                                                                                  1. Z-Distribution (μ=0, σ=1)
                                                                                                                                                                                                                    1. Standardization Process
                                                                                                                                                                                                                      1. Z-Score Calculation
                                                                                                                                                                                                                        1. Z-Table Usage
                                                                                                                                                                                                                          1. Reading Probabilities
                                                                                                                                                                                                                            1. Finding Critical Values
                                                                                                                                                                                                                              1. Interpolation Methods
                                                                                                                                                                                                                            2. The 68-95-99.7 Rule
                                                                                                                                                                                                                              1. Empirical Rule Statement
                                                                                                                                                                                                                                1. One Standard Deviation (68%)
                                                                                                                                                                                                                                  1. Two Standard Deviations (95%)
                                                                                                                                                                                                                                    1. Three Standard Deviations (99.7%)
                                                                                                                                                                                                                                      1. Applications in Quality Control
                                                                                                                                                                                                                                        1. Outlier Detection Guidelines
                                                                                                                                                                                                                                        2. Applications
                                                                                                                                                                                                                                          1. Natural Phenomena Modeling
                                                                                                                                                                                                                                            1. Measurement Errors
                                                                                                                                                                                                                                              1. Statistical Inference Foundation
                                                                                                                                                                                                                                            2. Exponential Distribution
                                                                                                                                                                                                                                              1. Definition and Context
                                                                                                                                                                                                                                                1. Continuous Analog of Geometric
                                                                                                                                                                                                                                                  1. Rate Parameter λ
                                                                                                                                                                                                                                                  2. Probability Density Function
                                                                                                                                                                                                                                                    1. λe^(-λx) Formula
                                                                                                                                                                                                                                                      1. Decreasing Function
                                                                                                                                                                                                                                                      2. Cumulative Distribution Function
                                                                                                                                                                                                                                                        1. 1 - e^(-λx) Formula
                                                                                                                                                                                                                                                        2. Properties
                                                                                                                                                                                                                                                          1. Mean (1/λ)
                                                                                                                                                                                                                                                            1. Variance (1/λ²)
                                                                                                                                                                                                                                                              1. Memoryless Property
                                                                                                                                                                                                                                                                1. Mathematical Expression
                                                                                                                                                                                                                                                                  1. Practical Implications
                                                                                                                                                                                                                                                                2. Applications
                                                                                                                                                                                                                                                                  1. Reliability Engineering
                                                                                                                                                                                                                                                                    1. Queuing Theory
                                                                                                                                                                                                                                                                      1. Survival Analysis
                                                                                                                                                                                                                                                                    2. t-Distribution (Student's t)
                                                                                                                                                                                                                                                                      1. Definition and Context
                                                                                                                                                                                                                                                                        1. Small Sample Inference
                                                                                                                                                                                                                                                                          1. Unknown Population Variance
                                                                                                                                                                                                                                                                          2. Degrees of Freedom Parameter
                                                                                                                                                                                                                                                                            1. Relationship to Sample Size
                                                                                                                                                                                                                                                                              1. Effect on Distribution Shape
                                                                                                                                                                                                                                                                              2. Properties
                                                                                                                                                                                                                                                                                1. Symmetry About Zero
                                                                                                                                                                                                                                                                                  1. Heavier Tails than Normal
                                                                                                                                                                                                                                                                                    1. Convergence to Normal
                                                                                                                                                                                                                                                                                    2. Comparison with Normal Distribution
                                                                                                                                                                                                                                                                                      1. Shape Differences
                                                                                                                                                                                                                                                                                        1. Tail Behavior
                                                                                                                                                                                                                                                                                          1. When to Use Each
                                                                                                                                                                                                                                                                                          2. Applications
                                                                                                                                                                                                                                                                                            1. Confidence Intervals
                                                                                                                                                                                                                                                                                              1. Hypothesis Testing
                                                                                                                                                                                                                                                                                                1. Small Sample Problems
                                                                                                                                                                                                                                                                                              2. Chi-Square Distribution
                                                                                                                                                                                                                                                                                                1. Definition and Context
                                                                                                                                                                                                                                                                                                  1. Degrees of Freedom
                                                                                                                                                                                                                                                                                                    1. Properties and Shape
                                                                                                                                                                                                                                                                                                      1. Applications in Testing
                                                                                                                                                                                                                                                                                                      2. F-Distribution
                                                                                                                                                                                                                                                                                                        1. Definition and Context
                                                                                                                                                                                                                                                                                                          1. Two Degrees of Freedom Parameters
                                                                                                                                                                                                                                                                                                            1. Applications in ANOVA
                                                                                                                                                                                                                                                                                                          2. The Central Limit Theorem (CLT)
                                                                                                                                                                                                                                                                                                            1. Statement and Mathematical Formulation
                                                                                                                                                                                                                                                                                                              1. Sample Mean Distribution
                                                                                                                                                                                                                                                                                                                1. Standardization Formula
                                                                                                                                                                                                                                                                                                                  1. Convergence to Normality
                                                                                                                                                                                                                                                                                                                  2. Conditions and Assumptions
                                                                                                                                                                                                                                                                                                                    1. Independent Observations
                                                                                                                                                                                                                                                                                                                      1. Identically Distributed
                                                                                                                                                                                                                                                                                                                        1. Finite Variance Requirement
                                                                                                                                                                                                                                                                                                                        2. Intuition and Explanation
                                                                                                                                                                                                                                                                                                                          1. Averaging Effect
                                                                                                                                                                                                                                                                                                                            1. Cancellation of Deviations
                                                                                                                                                                                                                                                                                                                              1. Universal Phenomenon
                                                                                                                                                                                                                                                                                                                              2. Sample Size Considerations
                                                                                                                                                                                                                                                                                                                                1. Rule of Thumb (n ≥ 30)
                                                                                                                                                                                                                                                                                                                                  1. Population Distribution Impact
                                                                                                                                                                                                                                                                                                                                    1. Skewness and Required Sample Size
                                                                                                                                                                                                                                                                                                                                    2. Significance and Applications
                                                                                                                                                                                                                                                                                                                                      1. Foundation of Inferential Statistics
                                                                                                                                                                                                                                                                                                                                        1. Confidence Interval Construction
                                                                                                                                                                                                                                                                                                                                          1. Hypothesis Testing Basis
                                                                                                                                                                                                                                                                                                                                            1. Quality Control Applications
                                                                                                                                                                                                                                                                                                                                            2. Sampling Distribution of the Mean
                                                                                                                                                                                                                                                                                                                                              1. Mean of Sample Means
                                                                                                                                                                                                                                                                                                                                                1. Standard Error Formula
                                                                                                                                                                                                                                                                                                                                                  1. Relationship to Population Parameters
                                                                                                                                                                                                                                                                                                                                                  2. Law of Large Numbers
                                                                                                                                                                                                                                                                                                                                                    1. Weak Law of Large Numbers
                                                                                                                                                                                                                                                                                                                                                      1. Strong Law of Large Numbers
                                                                                                                                                                                                                                                                                                                                                        1. Relationship to CLT
                                                                                                                                                                                                                                                                                                                                                          1. Practical Implications
                                                                                                                                                                                                                                                                                                                                                          2. Extensions and Variations
                                                                                                                                                                                                                                                                                                                                                            1. CLT for Proportions
                                                                                                                                                                                                                                                                                                                                                              1. CLT for Other Statistics
                                                                                                                                                                                                                                                                                                                                                                1. Finite Population Correction

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                                                                                                                                                                                                                                                                                                                                                            Go to top

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                                                                                                                                                                                                                                                                                                                                                            5. Inferential Statistics: From Samples to Populations