Statistics

1. Probability Theory
   1. Probability Concepts
      1. Experiment, Outcome, and Event
         1. Definitions
            1. Experiment: a procedure that yields random outcomes.
            2. Outcome: a single result of an experiment.
            3. Event: a set of outcomes of an experiment.
         2. Sample Space
            1. Definition and examples.
            2. Finite vs. Infinite sample spaces.
         3. Types of Events
            1. Simple and Compound events.
            2. Independent and Dependent events.
            3. Mutually Exclusive and Inclusive events.
      2. Probability Distributions
         1. Definitions
            1. Probability Mass Function (PMF) for discrete variables.
            2. Probability Density Function (PDF) for continuous variables.
         2. Cumulative Distribution Function (CDF)
            1. Definition and properties.
            2. Examples with graphs.
      3. Conditional Probability
         1. Definition and formula.
         2. Properties
            1. Multiplication rule.
            2. Independence assessment.
         3. Applications
            1. Medical testing.
            2. Risk assessment.
      4. Bayes' Theorem
         1. Formula and definitions.
         2. Interpretation and significance.
         3. Applications
            1. Spam filtering.
            2. Diagnostic test analysis.
   2. Discrete Distributions
      1. Binomial Distribution
         1. Properties
            1. Probability of success and failure.
            2. Mean and variance.
         2. Applications
            1. Quality control.
            2. Genetics.
      2. Poisson Distribution
         1. Properties
            1. Mean equals variance.
            2. Skewness.
         2. Applications
            1. Queueing theory.
            2. Rare event modeling.
      3. Geometric Distribution
         1. Properties
            1. Memoryless property.
            2. Mean and variance.
         2. Applications
            1. Modeling the number of trials until first success.
   3. Continuous Distributions
      1. Normal Distribution
         1. Properties
            1. Symmetry and shape.
            2. Standard normal distribution.
         2. Applications
            1. Natural phenomena modeling.
            2. Central Limit Theorem relevance.
      2. t-distribution
         1. Properties
            1. Heavier tails than normal distribution.
            2. Dependence on degrees of freedom.
         2. Applications
            1. Small sample hypothesis testing.
            2. Confidence intervals.
      3. Exponential Distribution
         1. Properties
            1. Memoryless property.
            2. Mean and variance.
         2. Applications
            1. Survival analysis.
            2. Time until an event occurs (e.g., failure time).
      4. Uniform Distribution
         1. Properties
            1. Constant probability.
            2. Bounded interval.
         2. Applications
            1. Random number generation.
            2. Simulations.
   4. Central Limit Theorem
      1. Concepts
         1. Definition and importance.
         2. Conditions for applicability.
      2. Implications
         1. Approximation of distributions.
         2. Importance in inferential statistics.
      3. Examples
         1. Empirical demonstration using sample means.
         2. Applications in hypothesis testing.