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Statistics
Probability Theory and Distributions
1. Foundational Concepts of Probability
2. Conditional Probability and Independence
3. Random Variables and Their Properties
4. Expectation and Moments
5. Common Discrete Distributions
6. Common Continuous Distributions
7. Joint Distributions and Dependence
8. Transformations and Functions of Random Variables
9. Limit Theorems and Convergence
Random Variables and Their Properties
Introduction to Random Variables
Definition as Functions
Mapping from Sample Space to Real Line
Measurability Conditions
Notation and Conventions
Motivation and Interpretation
Numerical Representation of Outcomes
Facilitating Mathematical Analysis
Types of Random Variables
Discrete Random Variables
Continuous Random Variables
Mixed Random Variables
Singular Random Variables
Discrete Random Variables
Characterization
Countable Range
Point Mass Distributions
Probability Mass Function
Definition and Properties
Validity Conditions
Graphical Representation
Cumulative Distribution Function
Definition for Discrete Variables
Step Function Properties
Right Continuity
Relationship to PMF
Functions of Discrete Random Variables
Transformation of PMF
One-to-One Transformations
Many-to-One Transformations
Continuous Random Variables
Characterization
Uncountable Range
Absolutely Continuous Distributions
Probability Density Function
Definition and Properties
Validity Conditions
Interpretation as Density
Graphical Representation
Cumulative Distribution Function
Definition for Continuous Variables
Continuity Properties
Relationship to PDF
Fundamental Theorem of Calculus Application
Functions of Continuous Random Variables
Change of Variables Technique
Jacobian Method
Inverse Transform Method
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2. Conditional Probability and Independence
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4. Expectation and Moments