Probabilistic Programming and Data Structures

Probabilistic Programming and Data Structures is an area of computer science that integrates probability theory directly into the design of algorithms and software. Probabilistic programming allows developers to create models that explicitly represent uncertainty, where variables are treated as probability distributions rather than fixed values, enabling sophisticated statistical inference and machine learning applications. Complementing this paradigm, probabilistic data structures, such as Bloom filters, HyperLogLog, and Count-Min Sketch, use randomization and hashing to provide approximate answers to queries about large datasets with a mathematically guaranteed level of accuracy, trading perfect precision for dramatic gains in memory efficiency and computational speed.

1. Foundational Concepts in Probability and Statistics
   1. Core Probability Theory
      1. Sample Spaces and Events
         1. Sample Space Definition
         2. Event Definition and Types
            1. Simple Events
            2. Compound Events
            3. Mutually Exclusive Events
            4. Exhaustive Events
         3. Event Operations
            1. Union of Events
            2. Intersection of Events
            3. Complement of Events
            4. De Morgan's Laws
      2. Axioms of Probability
         1. Non-negativity Axiom
         2. Normalization Axiom
         3. Additivity Axiom
         4. Consequences and Derived Properties
      3. Conditional Probability and Independence
         1. Conditional Probability Definition
         2. Multiplication Rule
         3. Law of Total Probability
         4. Independent Events
            1. Definition of Independence
            2. Pairwise Independence
            3. Mutual Independence
         5. Conditional Independence
      4. Bayes' Theorem
         1. Statement and Derivation
         2. Prior Probability
         3. Likelihood Function
         4. Posterior Probability
         5. Marginal Likelihood
         6. Applications in Inference
   2. Random Variables and Distributions
      1. Discrete Random Variables
         1. Definition and Properties
         2. Probability Mass Function
            1. Properties of PMF
            2. Support of Discrete Variables
         3. Cumulative Distribution Function for Discrete Variables
         4. Common Discrete Distributions
            1. Bernoulli Distribution
               1. Definition and Parameters
               2. Mean and Variance
            2. Binomial Distribution
               1. Definition and Parameters
               2. Mean and Variance
               3. Relationship to Bernoulli
            3. Poisson Distribution
               1. Definition and Parameters
               2. Mean and Variance
               3. Poisson Process
            4. Geometric Distribution
               1. Definition and Parameters
               2. Mean and Variance
               3. Memoryless Property
            5. Negative Binomial Distribution
               1. Definition and Parameters
               2. Mean and Variance
      2. Continuous Random Variables
         1. Definition and Properties
         2. Probability Density Function
            1. Properties of PDF
            2. Support of Continuous Variables
         3. Cumulative Distribution Function
            1. Properties of CDF
            2. Relationship between PDF and CDF
         4. Common Continuous Distributions
            1. Uniform Distribution
               1. Definition and Parameters
               2. Mean and Variance
            2. Normal Distribution
               1. Definition and Parameters
               2. Standard Normal Distribution
               3. Mean and Variance
               4. Central Role in Statistics
            3. Exponential Distribution
               1. Definition and Parameters
               2. Memoryless Property
               3. Mean and Variance
               4. Connection to Poisson Process
            4. Gamma Distribution
               1. Definition and Parameters
               2. Mean and Variance
               3. Relationship to Exponential
            5. Beta Distribution
               1. Definition and Parameters
               2. Mean and Variance
               3. Conjugate Prior Properties
   3. Properties of Random Variables
      1. Expectation
         1. Definition for Discrete Variables
         2. Definition for Continuous Variables
         3. Linearity of Expectation
         4. Expectation of Functions of Random Variables
      2. Variance and Standard Deviation
         1. Definition and Calculation
         2. Properties of Variance
         3. Variance of Linear Combinations
      3. Moments and Moment Generating Functions
         1. Raw Moments
         2. Central Moments
         3. Moment Generating Function
         4. Characteristic Function
      4. Covariance and Correlation
         1. Covariance Definition
         2. Properties of Covariance
         3. Correlation Coefficient
         4. Independence and Correlation
   4. Joint and Marginal Distributions
      1. Joint Probability Distributions
         1. Joint PMF for Discrete Variables
         2. Joint PDF for Continuous Variables
         3. Joint CDF
      2. Marginal Distributions
         1. Marginalization for Discrete Variables
         2. Marginalization for Continuous Variables
      3. Conditional Distributions
         1. Conditional PMF
         2. Conditional PDF
         3. Conditional Expectation
      4. Independence of Random Variables
         1. Definition of Independence
         2. Testing for Independence
   5. Limit Theorems
      1. Law of Large Numbers
         1. Weak Law of Large Numbers
         2. Strong Law of Large Numbers
         3. Convergence in Probability
         4. Almost Sure Convergence
      2. Central Limit Theorem
         1. Statement and Conditions
         2. Standardization
         3. Normal Approximation
         4. Applications in Sampling
   6. Statistical Inference Fundamentals
      1. Point Estimation
         1. Estimators and Estimates
         2. Properties of Estimators
            1. Unbiasedness
            2. Consistency
            3. Efficiency
            4. Sufficiency
         3. Method of Moments
         4. Maximum Likelihood Estimation
      2. Interval Estimation
         1. Confidence Intervals
         2. Construction Methods
         3. Interpretation of Confidence Level
      3. Hypothesis Testing
         1. Null and Alternative Hypotheses
         2. Test Statistics
         3. Critical Regions
         4. p-values
         5. Type I and Type II Errors
         6. Power of a Test
         7. Multiple Testing Corrections