Actuarial Science

2.

Mathematical and Statistical Foundations

2.1.

Probability Theory

2.1.1.

Fundamental Concepts

2.1.1.1.

Sample Spaces and Events

2.1.1.1.1.

Definition of Sample Space

2.1.1.1.2.

Types of Events

2.1.1.1.3.

Event Operations

2.1.1.1.4.

2.1.1.2.

Axioms of Probability

2.1.1.2.1.

Kolmogorov Axioms

2.1.1.2.2.

Properties Derived from Axioms

2.1.1.2.3.

Probability Measures

2.1.1.2.4.

Finite and Countable Spaces

2.1.1.3.

Set Theory in Probability

2.1.1.3.1.

Union and Intersection

2.1.1.3.2.

Complement Operations

2.1.1.3.3.

De Morgan's Laws

2.1.1.3.4.

2.1.1.4.

Probability Rules

2.1.1.4.1.

2.1.1.4.2.

Multiplication Rule

2.1.1.4.3.

Complement Rule

2.1.1.4.4.

Inclusion-Exclusion Principle

2.1.2.

Combinatorial Principles

2.1.2.1.

Counting Principles

2.1.2.1.1.

Fundamental Counting Principle

2.1.2.1.2.

Addition Principle

2.1.2.1.3.

Multiplication Principle

2.1.2.1.4.

Pigeonhole Principle

2.1.2.2.

2.1.2.2.1.

Permutations without Repetition

2.1.2.2.2.

Permutations with Repetition

2.1.2.2.3.

Circular Permutations

2.1.2.2.4.

Restricted Permutations

2.1.2.3.

2.1.2.3.1.

Combinations without Repetition

2.1.2.3.2.

Combinations with Repetition

2.1.2.3.3.

Pascal's Triangle

2.1.2.3.4.

Binomial Coefficients

2.1.2.4.

Multinomial Coefficients

2.1.2.4.1.

Definition and Properties

2.1.2.4.2.

Relationship to Binomial Coefficients

2.1.2.4.3.

Computational Methods

2.1.3.

Conditional Probability and Independence

2.1.3.1.

Conditional Probability

2.1.3.1.1.

Definition and Interpretation

2.1.3.1.2.

Properties of Conditional Probability

2.1.3.1.3.

Conditional Probability as Probability Measure

2.1.3.1.4.

Geometric Interpretation

2.1.3.2.

Multiplication Rule

2.1.3.2.1.

General Multiplication Rule

2.1.3.2.2.

2.1.3.2.3.

2.1.3.3.

Law of Total Probability

2.1.3.3.1.

Partition of Sample Space

2.1.3.3.2.

Formula and Applications

2.1.3.3.3.

Continuous Version

2.1.3.3.4.

Multiple Partitions

2.1.3.4.

2.1.3.4.1.

Statement and Proof

2.1.3.4.2.

Prior and Posterior Probabilities

2.1.3.4.3.

Applications in Actuarial Science

2.1.3.4.4.

Bayesian Updating

2.1.3.5.

Independence of Events

2.1.3.5.1.

Definition of Independence

2.1.3.5.2.

Mutual Independence

2.1.3.5.3.

Pairwise Independence

2.1.3.5.4.

Conditional Independence

2.1.4.

Random Variables

2.1.4.1.

Discrete Random Variables

2.1.4.1.1.

Definition and Examples

2.1.4.1.2.

Range and Support

2.1.4.1.3.

Probability Mass Function

2.1.4.1.4.

Properties and Characteristics

2.1.4.2.

Continuous Random Variables

2.1.4.2.1.

Definition and Examples

2.1.4.2.2.

Probability Density Function

2.1.4.2.3.

Relationship to Cumulative Distribution Function

2.1.4.2.4.

Properties and Characteristics

2.1.4.3.

Mixed Random Variables

2.1.4.3.1.

Definition and Examples

2.1.4.3.2.

Probability Function

2.1.4.3.3.

2.1.4.4.

Indicator Variables

2.1.4.4.1.

Definition and Properties

2.1.4.4.2.

Applications in Probability

2.1.4.4.3.

Expectation and Variance

2.1.4.4.4.

Combinations of Indicators

2.1.5.

Probability Distributions

2.1.5.1.

Probability Mass Function

2.1.5.1.1.

Definition and Properties

2.1.5.1.2.

Normalization Condition

2.1.5.1.3.

Relationship to CDF

2.1.5.1.4.

Examples and Applications

2.1.5.2.

Probability Density Function

2.1.5.2.1.

Definition and Properties

2.1.5.2.2.

Normalization Condition

2.1.5.2.3.

Relationship to CDF

2.1.5.2.4.

Examples and Applications

2.1.5.3.

Cumulative Distribution Function

2.1.5.3.1.

Definition and Properties

2.1.5.3.2.

Right Continuity

2.1.5.3.3.

Inverse Function

2.1.5.3.4.

Empirical Distribution Function

2.1.5.4.

Quantile Function

2.1.5.4.1.

Definition and Properties

2.1.5.4.2.

Percentiles and Quartiles

2.1.5.4.3.

2.1.5.4.4.

Applications in Risk Management

2.1.6.

Expectation and Moments

2.1.6.1.

2.1.6.1.1.

Definition for Discrete Variables

2.1.6.1.2.

Definition for Continuous Variables

2.1.6.1.3.

Properties of Expectation

2.1.6.1.4.

Linearity of Expectation

2.1.6.2.

Variance and Standard Deviation

2.1.6.2.1.

Definition and Calculation

2.1.6.2.2.

Properties of Variance

2.1.6.2.3.

Computational Formulas

2.1.6.2.4.

Coefficient of Variation

2.1.6.3.

Covariance and Correlation

2.1.6.3.1.

Definition of Covariance

2.1.6.3.2.

Properties of Covariance

2.1.6.3.3.

Correlation Coefficient

2.1.6.3.4.

Linear Relationships

2.1.6.4.

Higher-Order Moments

2.1.6.4.1.

2.1.6.4.2.

Central Moments

2.1.6.4.3.

Standardized Moments

2.1.6.4.4.

Moment Relationships

2.1.6.5.

Skewness and Kurtosis

2.1.6.5.1.

Measures of Asymmetry

2.1.6.5.2.

Measures of Tail Behavior

2.1.6.5.3.

Interpretation and Applications

2.1.6.5.4.

Sample Estimates

2.1.6.6.

Moment Generating Functions

2.1.6.6.1.

Definition and Properties

2.1.6.6.2.

Uniqueness Theorem

2.1.6.6.3.

Applications to Distribution Theory

2.1.6.6.4.

Cumulant Generating Functions

2.1.6.7.

Properties of Expectation

2.1.6.7.1.

Expectation of Functions

2.1.6.7.2.

Jensen's Inequality

2.1.6.7.3.

Conditional Expectation

2.1.6.7.4.

Law of Total Expectation

2.1.7.

Common Probability Distributions

2.1.7.1.

Discrete Distributions

2.1.7.1.1.

Bernoulli Distribution

2.1.7.1.1.1.

Definition and Parameters

2.1.7.1.1.2.

Moments and Properties

2.1.7.1.1.3.

Relationship to Other Distributions

2.1.7.1.2.

Binomial Distribution

2.1.7.1.2.1.

Definition and Parameters

2.1.7.1.2.2.

Moments and Properties

2.1.7.1.2.3.

Normal Approximation

2.1.7.1.2.4.

Applications in Insurance

2.1.7.1.3.

Poisson Distribution

2.1.7.1.3.1.

Definition and Parameters

2.1.7.1.3.2.

Moments and Properties

2.1.7.1.3.3.

Poisson Process

2.1.7.1.3.4.

Applications in Claim Modeling

2.1.7.1.4.

Geometric Distribution

2.1.7.1.4.1.

Definition and Parameters

2.1.7.1.4.2.

Memoryless Property

2.1.7.1.4.3.

Moments and Properties

2.1.7.1.5.

Negative Binomial Distribution

2.1.7.1.5.1.

Definition and Parameters

2.1.7.1.5.2.

Relationship to Geometric

2.1.7.1.5.3.

Moments and Properties

2.1.7.1.5.4.

Applications in Frequency Modeling

2.1.7.1.6.

Hypergeometric Distribution

2.1.7.1.6.1.

Definition and Parameters

2.1.7.1.6.2.

Sampling without Replacement

2.1.7.1.6.3.

Moments and Properties

2.1.7.1.7.

Multinomial Distribution

2.1.7.1.7.1.

Definition and Parameters

2.1.7.1.7.2.

Relationship to Binomial

2.1.7.2.

Continuous Distributions

2.1.7.2.1.

Uniform Distribution

2.1.7.2.1.1.

Definition and Parameters

2.1.7.2.1.2.

Moments and Properties

2.1.7.2.1.3.

Random Number Generation

2.1.7.2.2.

Normal Distribution

2.1.7.2.2.1.

Definition and Parameters

2.1.7.2.2.2.

Standard Normal Distribution

2.1.7.2.2.3.

Properties and Characteristics

2.1.7.2.2.4.

Central Limit Theorem Applications

2.1.7.2.3.

Exponential Distribution

2.1.7.2.3.1.

Definition and Parameters

2.1.7.2.3.2.

Memoryless Property

2.1.7.2.3.3.

Relationship to Poisson Process

2.1.7.2.3.4.

Applications in Survival Analysis

2.1.7.2.4.

Gamma Distribution

2.1.7.2.4.1.

Definition and Parameters

2.1.7.2.4.2.

2.1.7.2.4.3.

Moments and Properties

2.1.7.2.4.4.

Applications in Loss Modeling

2.1.7.2.5.

Beta Distribution

2.1.7.2.5.1.

Definition and Parameters

2.1.7.2.5.2.

Moments and Properties

2.1.7.2.5.3.

Applications in Bayesian Analysis

2.1.7.2.5.4.

Relationship to Other Distributions

2.1.7.2.6.

Lognormal Distribution

2.1.7.2.6.1.

Definition and Parameters

2.1.7.2.6.2.

Relationship to Normal Distribution

2.1.7.2.6.3.

Moments and Properties

2.1.7.2.6.4.

Applications in Financial Modeling

2.1.7.2.7.

Pareto Distribution

2.1.7.2.7.1.

Definition and Parameters

2.1.7.2.7.2.

Heavy-Tailed Properties

2.1.7.2.7.3.

Moments and Properties

2.1.7.2.7.4.

Applications in Loss Modeling

2.1.7.2.8.

Weibull Distribution

2.1.7.2.8.1.

Definition and Parameters

2.1.7.2.8.2.

Hazard Function

2.1.7.2.8.3.

Moments and Properties

2.1.7.2.8.4.

Applications in Reliability

2.1.8.

Joint, Marginal, and Conditional Distributions

2.1.8.1.

Joint Distributions

2.1.8.1.1.

Joint Probability Mass Function

2.1.8.1.2.

Joint Probability Density Function

2.1.8.1.3.

Joint Cumulative Distribution Function

2.1.8.1.4.

Properties and Relationships

2.1.8.2.

Marginal Distributions

2.1.8.2.1.

Marginal PMF and PDF

2.1.8.2.2.

Calculation Methods

2.1.8.2.3.

Relationship to Joint Distribution

2.1.8.3.

Conditional Distributions

2.1.8.3.1.

Conditional PMF and PDF

2.1.8.3.2.

Conditional Expectation

2.1.8.4.

Independence of Random Variables

2.1.8.4.1.

Definition and Tests

2.1.8.4.2.

Properties of Independent Variables

2.1.8.4.3.

Sums of Independent Variables

2.1.8.4.4.

Products of Independent Variables

2.1.8.5.

Covariance and Correlation of Random Variables

2.1.8.5.1.

Calculation Methods

2.1.8.5.2.

2.1.8.5.3.

Portfolio Applications

2.1.9.

Functions of Random Variables

2.1.9.1.

Distribution of Functions of Random Variables

2.1.9.1.1.

Transformation Method

2.1.9.1.2.

2.1.9.1.3.

Moment Generating Function Method

2.1.9.1.4.

2.1.9.2.

Change of Variables Technique

2.1.9.2.1.

One-to-One Transformations

2.1.9.2.2.

Many-to-One Transformations

2.1.9.2.3.

Jacobian Method

2.1.9.3.

2.1.9.3.1.

Definition and Properties

2.1.9.3.2.

Discrete Convolution

2.1.9.3.3.

Continuous Convolution

2.1.9.3.4.

Applications to Sums

2.1.10.

2.1.10.1.

Law of Large Numbers

2.1.10.1.1.

Weak Law of Large Numbers

2.1.10.1.1.1.

Statement and Conditions

2.1.10.1.1.2.

2.1.10.1.1.3.

Convergence in Probability

2.1.10.1.2.

Strong Law of Large Numbers

2.1.10.1.2.1.

Statement and Conditions

2.1.10.1.2.2.

Almost Sure Convergence

2.1.10.1.2.3.

Relationship to Weak Law

2.1.10.2.

Central Limit Theorem

2.1.10.2.1.

Classical Central Limit Theorem

2.1.10.2.2.

Lindeberg-Lévy Theorem

2.1.10.2.3.

Lyapunov Theorem

2.1.10.2.4.

Berry-Esseen Theorem

2.1.10.3.

Applications in Actuarial Science

2.1.10.3.1.

2.1.10.3.2.

Premium Calculation

2.1.10.3.3.

Solvency Assessment

2.1.10.3.4.

Simulation Studies

2.2.

Statistics and Data Analysis

2.2.1.

Descriptive Statistics

2.2.1.1.

Measures of Central Tendency

2.2.1.1.1.

2.2.1.1.2.

2.2.1.1.3.

2.2.1.1.4.

2.2.1.2.

Measures of Variability

2.2.1.2.1.

2.2.1.2.2.

2.2.1.2.3.

Standard Deviation

2.2.1.2.4.

Interquartile Range

2.2.1.3.

Measures of Shape

2.2.1.3.1.

2.2.1.3.2.

2.2.1.3.3.

2.2.1.3.4.

2.2.1.4.

Data Visualization

2.2.1.4.1.

2.2.1.4.2.

Time Series Plots

2.2.1.4.3.

Probability Plots

2.2.1.4.4.

Quantile-Quantile Plots

2.2.2.

Statistical Inference

2.2.2.1.

Point Estimation

2.2.2.1.1.

Method of Moments

2.2.2.1.2.

Maximum Likelihood Estimation

2.2.2.1.3.

Properties of Estimators

2.2.2.1.4.

Bias and Efficiency

2.2.2.2.

Interval Estimation

2.2.2.2.1.

Confidence Intervals

2.2.2.2.2.

Bootstrap Methods

2.2.2.2.3.

Bayesian Credible Intervals

2.2.2.2.4.

Prediction Intervals

2.2.2.3.

Hypothesis Testing

2.2.2.3.1.

Null and Alternative Hypotheses

2.2.2.3.2.

Type I and Type II Errors

2.2.2.3.3.

Test Statistics

2.2.2.3.4.

P-values and Critical Values

2.2.2.4.

Goodness of Fit Tests

2.2.2.4.1.

Chi-Square Test

2.2.2.4.2.

Kolmogorov-Smirnov Test

2.2.2.4.3.

Anderson-Darling Test

2.2.2.4.4.

Likelihood Ratio Tests

2.2.3.

Regression Analysis

2.2.3.1.

Simple Linear Regression

2.2.3.1.1.

Model Assumptions

2.2.3.1.2.

Parameter Estimation

2.2.3.1.3.

Inference for Parameters

2.2.3.1.4.

Model Diagnostics

2.2.3.2.

Multiple Linear Regression

2.2.3.2.1.

Model Specification

2.2.3.2.2.

Parameter Estimation

2.2.3.2.3.

Model Selection

2.2.3.2.4.

Multicollinearity

2.2.3.3.

Generalized Linear Models

2.2.3.3.1.

Exponential Family

2.2.3.3.2.

2.2.3.3.3.

Parameter Estimation

2.2.3.3.4.

Model Diagnostics

2.2.3.4.

Time Series Analysis

2.2.3.4.1.

Trend and Seasonality

2.2.3.4.2.

Autoregressive Models

2.2.3.4.3.

Moving Average Models

2.2.3.4.4.

2.3.

Financial Mathematics

2.3.1.

Theory of Interest

2.3.1.1.

Time Value of Money

2.3.1.1.1.

Present Value Concept

2.3.1.1.2.

Future Value Concept

2.3.1.1.3.

Equivalence of Cash Flows

2.3.1.1.4.

Real vs. Nominal Interest Rates

2.3.1.2.

Interest Rate Measurement

2.3.1.2.1.

Simple Interest

2.3.1.2.1.1.

Definition and Formula

2.3.1.2.1.2.

2.3.1.2.1.3.

Comparison with Compound Interest

2.3.1.2.2.

Compound Interest

2.3.1.2.2.1.

Definition and Formula

2.3.1.2.2.2.

Frequency of Compounding

2.3.1.2.2.3.

Continuous Compounding

2.3.1.2.3.

Nominal vs. Effective Rates

2.3.1.2.3.1.

Nominal Interest Rate

2.3.1.2.3.2.

Effective Annual Rate

2.3.1.2.3.3.

Conversion Formulas

2.3.1.2.4.

Force of Interest

2.3.1.2.4.1.

Definition and Properties

2.3.1.2.4.2.

Relationship to Other Interest Measures

2.3.1.2.4.3.

Continuous Compounding

2.3.1.3.

Discounting and Accumulation

2.3.1.3.1.

Present Value Factors

2.3.1.3.2.

Accumulation Factors

2.3.1.3.3.

Discount Factors

2.3.1.3.4.

Relationship Between Factors

2.3.1.4.

Present and Accumulated Value Factors

2.3.1.4.1.

Single Payment Factors

2.3.1.4.2.

Series Payment Factors

2.3.1.4.3.

Perpetuity Factors

2.3.1.4.4.

Applications in Valuation

2.3.2.

2.3.2.1.

Annuity-Immediate and Annuity-Due

2.3.2.1.1.

Definitions and Notation

2.3.2.1.2.

Present Value Calculations

2.3.2.1.3.

Accumulated Value Calculations

2.3.2.1.4.

Relationship Between Types

2.3.2.2.

Level Payment Annuities

2.3.2.2.1.

Ordinary Annuities

2.3.2.2.2.

2.3.2.2.3.

Deferred Annuities

2.3.2.2.4.

2.3.2.3.

Varying Annuities

2.3.2.3.1.

Arithmetically Increasing Annuities

2.3.2.3.1.1.

Present Value Formulas

2.3.2.3.1.2.

Accumulated Value Formulas

2.3.2.3.1.3.

2.3.2.3.2.

Arithmetically Decreasing Annuities

2.3.2.3.2.1.

Present Value Formulas

2.3.2.3.2.2.

Accumulated Value Formulas

2.3.2.3.2.3.

2.3.2.3.3.

Geometrically Increasing Annuities

2.3.2.3.3.1.

Present Value Formulas

2.3.2.3.3.2.

Accumulated Value Formulas

2.3.2.3.3.3.

Inflation Adjustments

2.3.2.3.4.

Geometrically Decreasing Annuities

2.3.2.3.4.1.

Present Value Formulas

2.3.2.3.4.2.

Accumulated Value Formulas

2.3.2.3.4.3.

2.3.2.4.

Continuous Annuities

2.3.2.4.1.

Continuous Payment Streams

2.3.2.4.2.

Present Value Calculations

2.3.2.4.3.

Accumulated Value Calculations

2.3.2.5.

2.3.2.5.1.

Level Perpetuities

2.3.2.5.2.

Increasing Perpetuities

2.3.2.5.3.

Decreasing Perpetuities

2.3.2.6.

Deferred Annuities

2.3.2.6.1.

Deferred Ordinary Annuities

2.3.2.6.2.

Deferred Annuities Due

2.3.2.6.3.

Present Value Calculations

2.3.3.

Loans and Amortization

2.3.3.1.

Loan Structure and Terms

2.3.3.1.1.

Principal Amount

2.3.3.1.2.

2.3.3.1.3.

Repayment Period

2.3.3.1.4.

Payment Frequency

2.3.3.2.

Amortization Method

2.3.3.2.1.

Level Payment Loans

2.3.3.2.2.

Interest and Principal Components

2.3.3.2.3.

Amortization Schedules

2.3.3.2.4.

Outstanding Balance Calculations

2.3.3.3.

Sinking Fund Method

2.3.3.3.1.

Sinking Fund Concept

2.3.3.3.2.

Payment Calculations

2.3.3.3.3.

Fund Accumulation

2.3.3.3.4.

Comparison with Amortization

2.3.3.4.

2.3.3.4.1.

Payment Schedules

2.3.3.4.2.

Interest Schedules

2.3.3.4.3.

Principal Schedules

2.3.3.4.4.

Balance Schedules

2.3.3.5.

Outstanding Loan Balance

2.3.3.5.1.

Prospective Method

2.3.3.5.2.

Retrospective Method

2.3.3.5.3.

Refinancing Considerations

2.3.4.

Bonds and Other Securities

2.3.4.1.

2.3.4.1.1.

Present Value of Cash Flows

2.3.4.1.2.

Coupon Payments

2.3.4.1.3.

Principal Repayment

2.3.4.1.4.

Market Price Determination

2.3.4.2.

Yield to Maturity

2.3.4.2.1.

Definition and Calculation

2.3.4.2.2.

Relationship to Bond Price

2.3.4.2.3.

2.3.4.3.

Duration and Convexity

2.3.4.3.1.

Macaulay Duration

2.3.4.3.2.

Modified Duration

2.3.4.3.3.

Effective Duration

2.3.4.3.4.

Convexity Measures

2.3.4.4.

Callable and Putable Bonds

2.3.4.4.1.

2.3.4.4.2.

2.3.4.4.3.

Option-Adjusted Spreads

2.3.4.4.4.

Valuation Methods

2.3.4.5.

Price-Yield Relationship

2.3.4.5.1.

Inverse Relationship

2.3.4.5.2.

Price Sensitivity

2.3.4.5.3.

Yield Curve Risk

2.3.5.

Deterministic Cash Flow Analysis

2.3.5.1.

Net Present Value

2.3.5.1.1.

Definition and Calculation

2.3.5.1.2.

2.3.5.1.3.

2.3.5.2.

Internal Rate of Return

2.3.5.2.1.

Definition and Calculation

2.3.5.2.2.

2.3.5.2.3.

2.3.5.3.

Project Appraisal

2.3.5.3.1.

Investment Criteria

2.3.5.3.2.

Risk Assessment

2.3.5.3.3.

Sensitivity Analysis

2.3.5.3.4.

Decision Making

2.3.6.

Asset-Liability Management

2.3.6.1.

2.3.6.1.1.

Macaulay Duration

2.3.6.1.2.

Modified Duration

2.3.6.1.3.

Key Rate Duration

2.3.6.1.4.

Portfolio Duration

2.3.6.2.

2.3.6.2.1.

Definition and Calculation

2.3.6.2.2.

Positive and Negative Convexity

2.3.6.2.3.

Portfolio Convexity

2.3.6.3.

2.3.6.3.1.

Classical Immunization

2.3.6.3.2.

Contingent Immunization

2.3.6.3.3.

Horizon Matching

2.3.6.3.4.

Risk Minimization

2.3.6.4.

Matching Strategies

2.3.6.4.1.

Cash Flow Matching

2.3.6.4.2.

Duration Matching

2.3.6.4.3.

Convexity Matching

2.3.6.4.4.

Dynamic Strategies

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1. Foundations of Actuarial Science

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3. Life Contingencies and Survival Models