Artificial Intelligence with PHP
  • Getting Started
    • Introduction
    • Audience
    • How to Read This Book
    • Glossary
    • Contributors
    • Resources
    • Changelog
  • Artificial Intelligence
    • Introduction
    • Overview of AI
      • History of AI
      • How Does AI Work?
      • Structure of AI
      • Will AI Take Over the World?
      • Types of AI
        • Limited Memory AI
        • Reactive AI
        • Theory of Mind AI
        • Self-Aware AI
    • AI Capabilities in PHP
      • Introduction to LLM Agents PHP SDK
      • Overview of AI Libraries in PHP
    • AI Agents
      • Introduction to AI Agents
      • Structure of AI Agent
      • Components of AI Agents
      • Types of AI Agents
      • AI Agent Architecture
      • AI Agent Environment
      • Application of Agents in AI
      • Challenges in AI Agent Development
      • Future of AI Agents
      • Turing Test in AI
      • LLM AI Agents
        • Introduction to LLM AI Agents
        • Implementation in PHP
          • Sales Analyst Agent
          • Site Status Checker Agent
    • Theoretical Foundations of AI
      • Introduction to Theoretical Foundations of AI
      • Problem Solving in AI
        • Introduction
        • Types of Search Algorithms
          • Comparison of Search Algorithms
          • Informed (Heuristic) Search
            • Global Search
              • Beam Search
              • Greedy Search
              • Iterative Deepening A* Search
              • A* Search
                • A* Graph Search
                • A* Graph vs A* Tree Search
                • A* Tree Search
            • Local Search
              • Hill Climbing Algorithm
                • Introduction
                • Best Practices and Optimization
                • Practical Applications
                • Implementation in PHP
              • Simulated Annealing Search
              • Local Beam Search
              • Genetic Algorithms
              • Tabu Search
          • Uninformed (Blind) Search
            • Global Search
              • Bidirectional Search (BDS)
              • Breadth-First Search (BFS)
              • Depth-First Search (DFS)
              • Iterative Deepening Depth-First Search (IDDFS)
              • Uniform Cost Search (UCS)
            • Local Search
              • Depth-Limited Search (DLS)
              • Random Walk Search (RWS)
          • Adversarial Search
          • Means-Ends Analysis
      • Knowledge & Uncertainty in AI
        • Knowledge-Based Agents
        • Knowledge Representation
          • Introduction
          • Approaches to KR in AI
          • The KR Cycle in AI
          • Types of Knowledge in AI
          • KR Techniques
            • Logical Representation
              • Propositional Logic
              • Predicate Logic
              • Implementation in PHP
            • Semantic Network Representation
            • Frame Representation
            • Production Rules
        • Reasoning in AI
        • Uncertain Knowledge Representation
        • The Wumpus World
        • Applications and Challenges
      • Cybernetics and AI
      • Philosophical and Ethical Foundations of AI
    • Mathematics for AI
      • Computational Theory in AI
      • Logic and Reasoning
      • Set Theory and Discrete Mathematics
      • Decision Making in AI
    • Key Application of AI
      • AI in Astronomy
      • AI in Agriculture
      • AI in Automotive Industry
      • AI in Data Security
      • AI in Dating
      • AI in E-commerce
      • AI in Education
      • AI in Entertainment
      • AI in Finance
      • AI in Gaming
      • AI in Healthcare
      • AI in Robotics
      • AI in Social Media
      • AI in Software Development
      • AI in Adult Entertainment
      • AI in Criminal Justice
      • AI in Criminal World
      • AI in Military Domain
      • AI in Terrorist Activities
      • AI in Transforming Our World
      • AI in Travel and Transport
    • Practice
  • Machine Learning
    • Introduction
    • Overview of ML
      • History of ML
        • Origins and Early Concepts
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Terms and Principles
      • Machine Learning Life Cycle
      • Problems and Challenges
    • ML Capabilities in PHP
      • Overview of ML Libraries in PHP
      • Configuring an Environment for PHP
        • Direct Installation
        • Using Docker
        • Additional Notes
      • Introduction to PHP-ML
      • Introduction to Rubix ML
    • Mathematics for ML
      • Linear Algebra
        • Scalars
          • Definition and Operations
          • Scalars with PHP
        • Vectors
          • Definition and Operations
          • Vectors in Machine Learning
          • Vectors with PHP
        • Matrices
          • Definition and Types
          • Matrix Operations
          • Determinant of a Matrix
          • Inverse Matrices
          • Cofactor Matrices
          • Adjugate Matrices
          • Matrices in Machine Learning
          • Matrices with PHP
        • Tensors
          • Definition of Tensors
          • Tensor Properties
            • Tensor Types
            • Tensor Dimension
            • Tensor Rank
            • Tensor Shape
          • Tensor Operations
          • Practical Applications
          • Tensors in Machine Learning
          • Tensors with PHP
        • Linear Transformations
          • Introduction
          • LT with PHP
          • LT Role in Neural Networks
        • Eigenvalues and Eigenvectors
        • Norms and Distances
        • Linear Algebra in Optimization
      • Calculus
      • Probability and Statistics
      • Information Theory
      • Optimization Techniques
      • Graph Theory and Networks
      • Discrete Mathematics and Combinatorics
      • Advanced Topics
    • Data Fundamentals
      • Data Types and Formats
        • Data Types
        • Structured Data Formats
        • Unstructured Data Formats
        • Implementation with PHP
      • General Data Processing
        • Introduction
        • Storage and Management
          • Data Security and Privacy
          • Data Serialization and Deserialization in PHP
          • Data Versioning and Management
          • Database Systems for AI
          • Efficient Data Storage Techniques
          • Optimizing Data Retrieval for AI Algorithms
          • Big Data Considerations
            • Introduction
            • Big Data Techniques in PHP
      • ML Data Processing
        • Introduction
        • Types of Data in ML
        • Stages of Data Processing
          • Data Acquisition
            • Data Collection
            • Ethical Considerations in Data Preparation
          • Data Cleaning
            • Data Cleaning Examples
            • Data Cleaning Types
            • Implementation with PHP
          • Data Transformation
            • Data Transformation Examples
            • Data Transformation Types
            • Implementation with PHP ?..
          • Data Integration
          • Data Reduction
          • Data Validation and Testing
            • Data Splitting and Sampling
          • Data Representation
            • Data Structures in PHP
            • Data Visualization Techniques
          • Typical Problems with Data
    • ML Algorithms
      • Classification of ML Algorithms
        • By Methods Used
        • By Learning Types
        • By Tasks Resolved
        • By Feature Types
        • By Model Depth
      • Supervised Learning
        • Regression
          • Linear Regression
            • Types of Linear Regression
            • Finding Best Fit Line
            • Gradient Descent
            • Assumptions of Linear Regression
            • Evaluation Metrics for Linear Regression
            • How It Works by Math
            • Implementation in PHP
              • Multiple Linear Regression
              • Simple Linear Regression
          • Polynomial Regression
            • Introduction
            • Implementation in PHP
          • Support Vector Regression
        • Classification
        • Recommendation Systems
          • Matrix Factorization
          • User-Based Collaborative Filtering
      • Unsupervised Learning
        • Clustering
        • Dimension Reduction
        • Search and Optimization
        • Recommendation Systems
          • Item-Based Collaborative Filtering
          • Popularity-Based Recommendations
      • Semi-Supervised Learning
        • Regression
        • Classification
        • Clustering
      • Reinforcement Learning
      • Distributed Learning
    • Integrating ML into Web
      • Open-Source Projects
      • Introduction to EasyAI-PHP
    • Key Applications of ML
    • Practice
  • Neural Networks
    • Introduction
    • Overview of NN
      • History of NN
      • Basic Components of NN
        • Activation Functions
        • Connections and Weights
        • Inputs
        • Layers
        • Neurons
      • Problems and Challenges
      • How NN Works
    • NN Capabilities in PHP
    • Mathematics for NN
    • Types of NN
      • Classification of NN Types
      • Basic NN
        • Simple Perceptron
        • Implementation in PHP
          • Simple Perceptron with Libraries
          • Simple Perceptron with Pure PHP
      • NN with Hidden Layers
      • Deep Learning
      • Bayesian Neural Networks
      • Convolutional Neural Networks (CNN)
      • Recurrent Neural Networks (RNN)
    • Integrating NN into Web
    • Key Applications of NN
    • Practice
  • Natural Language Processing
    • Introduction
    • Overview of NLP
      • History of NLP
        • Ancient Times
        • Medieval Period
        • 15th-16th Century
        • 17th-18th Century
        • 19th Century
        • 20th Century
        • 21st Century
        • Coming Years
      • Key Concepts in NLP
      • Common Challenges in NLP
      • Machine Learning Role in NLP
    • NLP Capabilities in PHP
      • Overview of NLP Libraries in PHP
      • Challenges in NLP with PHP
    • Mathematics for NLP
    • NLP Techniques
      • Basic Text Processing with PHP
      • NLP Workflow
      • Popular Tools and Frameworks for NLP
      • Techniques and Algorithms in NLP
        • Basic NLP Techniques
        • Advanced NLP Techniques
      • Advanced NLP Topics
    • Integrating NLP into Web
    • Key Applications of NLP
    • Practice
  • Computer Vision
    • Introduction
  • Robotics
    • Introduction
  • Expert Systems
    • Introduction
    • Overview of ES
      • History of ES
        • Origins and Early ES
        • Milestones in the Evolution of ES
        • Expert Systems in Modern AI
      • Core Components and Architecture
      • Challenges and Limitations
      • Future Trends
    • ES Capabilities in PHP
    • Mathematics for ES
    • Building ES
      • Knowledge Representation Approaches
      • Inference Mechanisms
      • Best Practices for Knowledge Base Design and Inference
    • Integration ES into Web
    • Key Applications of ES
    • Practice
  • Cognitive Computing
    • Introduction
    • Overview of CC
      • History of CC
      • Differences Between CC and AI
    • CC Compatibilities in PHP
    • Mathematics for CC
    • Building CC
      • Practical Implementation
    • Integration CC into Web
    • Key Applications of CC
    • Practice
  • AI Ethics and Safety
    • Introduction
    • Overview of AI Ethics
      • Core Principles of AI Ethics
      • Responsible AI Development
      • Looking Ahead: Ethical AI Governance
    • Building Ethics & Safety AI
      • Fairness, Bias, and Transparency
        • Bias in AI Models
        • Model Transparency and Explainability
        • Auditing, Testing, and Continuous Monitoring
      • Privacy and Security in AI
        • Data Privacy and Consent
        • Safety Mechanisms in AI Integration
        • Preventing and Handling AI Misuse
      • Ensuring AI Accountability
        • Ethical AI in Decision Making
        • Regulations & Compliance
        • AI Risk Assessment
    • Key Applications of AI Ethics
    • Practice
  • Epilog
    • Summing-up
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Probability and Statistics

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Last updated 18 days ago

Basic Probability Concepts

  • Fundamentals of Probability

  • Discrete and Continuous Distributions

  • Joint and Marginal Probabilities

  • Probability spaces, events, and outcomes

  • Conditional probability and independence

  • Random Variables and Distributions, Expectations

  • Discrete and continuous random variables

  • Common probability distributions: Normal, Binomial, Poisson, etc.

  • Expectation and Variance

  • Mathematical Expectation and Algorithm Evaluation

  • Statistical Measures: Mean, Variance, and Standard Deviation

  • Covariance and correlation

  • Statistical Inference

  • Hypothesis testing and confidence intervals

  • Maximum likelihood estimation (MLE)\

  • Descriptive and inferential statistics applied to AI

    • Measures of Central Tendency and Dispersion

    • Data Visualization in AI

    • Statistical Inference for Supervised Learning

    • Hypothesis Testing in Validation of AI Models:

  • Conditional Probability and Bayes’ Theorem and its application in machine learning

    • Application in Bayesian Classification

    • Probabilistic Graphic Models

    • Bayesian Learning Algorithms

    • Bayes Theorem in Parameter Estimation

Probability and Statistics