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Chapter 2: Random Variables

  1. FRM® Part 1 Mastery Series
  2. Chapter 2: Random Variables
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Study Notes: Random Variables
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Interactive Practice Question Platform
Interactive Practice Question Platform
Tutorial – How to Use the Interactive Practice Question Platform
Direct Link – Interactive Practice Question Platform
Foundations of Risk Management
Introduction to Foundations of Risk Management
Instructional Video: Intro to Foundations of Risk
Learning Spreadsheet: Intro to VaR
Chapter 1: The Building Blocks of Risk Management
Study Notes: The Building Blocks of Risk Management
Practice Question Set: The Building Blocks of Risk Management
Instructional Video: The Building Blocks of Risk Management
Chapter 2: How Do Firms Manage Financial Risk?
Study Notes: How Do Firms Manage Financial Risk?
Practice Question Set: How Do Firms Manage Financial Risk?
Instructional Video: How Do Firms Manage Financial Risk?
Chapter 3: The Governance of Risk Management
Study Notes: The Governance of Risk Management
Practice Question Set: The Governance of Risk Management
Instructional Video: The Governance of Risk Management
Chapter 4. Credit Risk Transfer Mechanisms
Study Notes: Credit Risk Transfer Mechanisms
Practice Question Set: Credit Risk Transfer Mechanisms
Instructional Video: Credit Risk Transfer Mechanisms
Chapter 5: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Study Notes: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Practice Question Set: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Instructional Video: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Learning Spreadsheet 1 of 2: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Learning Spreadsheet 2 of 2: Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)
Chapter 6: The Arbitrage Pricing Theory and Multifactor Models of Risk and Return
Study Notes: APT and Multifactor Models of Risk and Return
Practice Question Set: APT and Multifactor Models of Risk and Return
Instructional Video: APT and Multifactor Models of Risk and Return
Learning Spreadsheet: APT and Multifactor Models of Risk and Return
Chapter 7: Principles for Effective Data Aggregation and Risk Reporting
Study Notes: Principles for Effective Data Aggregation and Risk Reporting
Practice Question Set: Risk Data Aggregation and Reporting Principles
Chapter 8: Enterprise Risk Management and Future Trends
Study Notes: Enterprise Risk Management (ERM) and Future Trends
Practice Question Set: ERM and Future Trends
Instructional Video: Enterprise Risk Management and Future Trends
Chapter 9: Learning from Financial Disasters
Study Notes: Learning from Financial Disasters
Practice Question Set: Learning from Financial Disasters
Instructional Video: Learning from Financial Disasters
Chapter 10: Anatomy of the Great Financial Crisis
Study Notes: Anatomy of the Great Financial Crisis
Practice Question Set: Anatomy of the Great Financial Crisis
Instructional Video: Anatomy of the Great Financial Crisis
Chapter 11: GARP Code of Conduct
Study Notes: GARP Code of Conduct
Practice Question Set: GARP Code of Conduct
Foundations of Risk Topic Review
Foundations of Risk Focus Review Video 1 of 2
Foundations of Risk Focus Review Video 2 of 2
Learning Spreadsheets: P1.T1.XLS Bundle
Quantitative Analysis
Chapter 1: Fundamentals of Probability
Study Notes: Fundamentals of Probability
Practice Question Set: Fundamentals of Probability
Instructional Video: Fundamentals of Probability
Learning Spreadsheet: Fundamentals of Probability
Chapter 2: Random Variables
Study Notes: Random Variables
Practice Question Set: Random Variables
Instructional Video: Random Variables
Learning Spreadsheet: Random Variables
Chapter 3: Common Univariate Random Variables
Study Notes: Common Univariate Random Variables
Practice Question Set: Common Univariate Random Variables
Instructional Video: Common Univariate Random Variables
Learning Spreadsheet: Common Univariate Random Variables
Chapter 4: Multivariate Random Variables
Study Notes: Multivariate Random Variables
Practice Question Set: Multivariate Random Variables
Instructional Video: Multivariate Random Variables
Learning Spreadsheet: Multivariate Random Variables
Chapter 5: Sample Moments
Study Notes: Sample Moments
Practice Question Set: Sample Moments
Instructional Video: Sample Moments
Learning Spreadsheet: Sample Moments
Chapter 6: Hypothesis Testing
Study Notes: Hypothesis Testing
Practice Question Set: Hypothesis Testing
Instructional Video: Hypothesis Testing
Learning Spreadsheet: Hypothesis Testing
Chapter 7: Linear Regression
Study Notes: Linear Regression
Practice Question Set: Linear Regression
Instructional Video: Linear Regression
Learning Spreadsheet: Linear Regression
Chapter 8: Regression with Multiple Explanatory Variables
Study Notes: Regression with Multiple Explanatory Variables
Practice Question Set: Regression with Multiple Explanatory Variables
Instructional Video 1 of 2: Regression with Multiple Explanatory Variables
Instructional Video 2 of 2: Regression with Multiple Explanatory Variables
Learning Spreadsheet: Regression with Multiple Explanatory Variables
Chapter 9: Regression Diagnostics
Study Notes: Regression Diagnostics
Practice Question Set: Regression Diagnostics
Instructional Video: Regression Diagnostics
Learning Spreadsheet: Regression Diagnostics
Chapter 10: Stationary Time Series
Study Notes: Stationary Time Series
Practice Question Set: Stationary Time Series
Instructional Video: Stationary Time Series
Learning Spreadsheet: Stationary Time Series
Chapter 11: Nonstationary Time Series
Study Notes: Nonstationary Time Series
Practice Question Set: Nonstationary Time Series
Instructional Video: Nonstationary Time Series
Learning Spreadsheet: Nonstationary Time Series
Chapter 12: Measuring Return, Volatility, and Correlation
Study Notes: Measuring Return, Volatility, and Correlation
Practice Question Set: Measuring Return, Volatility, and Correlation
Instructional Video: Chapter 12: Measuring Return, Volatility, and Correlation
Learning Spreadsheet: Measuring Return, Volatility, and Correlation
ARCHIVED: Instructional Video 1 of 2: Measuring Return, Volatility, and Correlation
ARCHIVED: Instructional Video 2 of 2: Measuring Return, Volatility, and Correlation
Chapter 13: Simulation and Bootstrapping
Study Notes: Simulation and Bootstrapping
Practice Question Set: Simulation and Bootstrapping
Instructional Video: Simulation and Bootstrapping
Learning Spreadsheet: Simulation and Bootstrapping
Archived Instructional Video: Simulation and Bootstrapping
Chapter 14: Machine-Learning Methods
Study Notes: Machine-Learning Methods
Practice Questions: Machine-Learning Methods
Instructional Video: Machine-Learning Methods
Learning Spreadsheet: Machine-Learning Methods
Chapter 15: Machine Learning and Prediction
Study Notes: Machine Learning and Prediction
Practice Questions: Machine Learning and Prediction
Instructional Video: Machine-Learning and Prediction
Learning Spreadsheet: Machine-Learning and Prediction
Quantitative Analysis Topic Review
Quantitative Focus Review Video (1 of 3)
Quantitative Focus Review Video (2 of 3)
Quantitative Focus Review Video (3 of 3)
Learning Spreadsheets: P1.T2.a XLS Bundle
Learning Spreadsheets: P1.T2.b XLS Bundle
Learning Spreadsheets: P1.T2.c XLS Bundle
Learning Spreadsheets: P1.T2.d XLS Bundle
Financial Markets & Products
Chapter 1: Banks
Study Notes: Banks
Practice Question Set: Banks
Instructional Video: Banks
Learning Spreadsheet: Banks
Chapter 2. Insurance Companies and Pension Plans
Study Notes: Insurance Companies and Pension Plans
Practice Question Set: Insurance Companies and Pension Plans
Instructional Video: Insurance Companies and Pension Plans
Learning Spreadsheet: Insurance Companies and Pension Plans
Chapter 3: Fund Management
Study Notes: Fund Management
Practice Question Set: Fund Management
Instructional Video: Fund Management
Learning Spreadsheet: Fund Management
Chapter 4. Introduction to Derivatives
Study Notes: Introduction to Derivatives
Practice Question Set: Introduction to Derivatives
Instructional Video: Introduction to Derivatives
Learning Spreadsheet: Introduction to Derivatives
Chapter 5: Exchanges and OTC Markets
Study Notes: Exchanges and OTC Markets
Practice Question Set: Exchanges and OTC Markets
Instructional Video: Chapter 5: Exchanges and OTC Markets & Chapter 6: Central Clearing
Learning Spreadsheet: Exchanges and OTC Markets
Chapter 6: Central Clearing
Study Notes: Central Clearing
Practice Question Set: Central Clearing
Learning Spreadsheet: Central Clearing
Instructional Video: See Chapter 5 Exchanged and OTC Markets (combined video)
Chapter 7: Futures Markets
Study Notes: Futures Markets
Practice Question Set: Futures Markets
Instructional Video: Futures Markets
Learning Spreadsheet: Futures Markets
Chapter 8: Using Futures for Hedging
Study Notes: Using Futures for Hedging
Practice Question Set: Using Futures for Hedging
Instructional Video: Using Futures for Hedging
Learning Spreadsheet: Using Futures for Hedging
Chapter 9: Foreign Exchange Markets
Study Notes: Foreign Exchange Markets
Practice Question Set: Foreign Exchange Markets
Instructional Video: Foreign Exchange Markets
Learning Spreadsheet: Foreign Exchange Markets
Chapter 10: Pricing Financial Forwards and Futures
Study Notes: Pricing Financial Forwards and Futures
Practice Question Set: Pricing Financial Forwards and Futures
Instructional Video: Pricing Financial Forwards and Futures
Learning Spreadsheet: Pricing Financial Forwards and Futures
Chapter 11: Commodity Forwards and Futures
Study Notes: Commodity Forwards and Futures
Practice Question Set: Commodity Forwards and Futures
Instructional Video: Commodity Forwards and Futures
Learning Spreadsheet: Commodity Forwards and Futures
Chapter 12: Options Markets
Study Notes: Options Markets
Practice Question Set: Options Markets
Instructional Video: Options Markets
Learning Spreadsheet: Options Markets
Chapter 13: Properties of Options
Study Notes: Properties of Options
Practice Question Set: Properties of Options
Instructional Video: Properties of Options
Learning Spreadsheet: Properties of Options
Chapter 14: Trading Strategies
Study Notes: Trading Strategies
Practice Question Set: Trading Strategies
Instructional Video: Trading Strategies
Learning Spreadsheet: Trading Strategies
Chapter 15: Exotic Options
Study Notes: Exotic Options
Practice Question Set: Exotic Options
Instructional Video: Exotic Options
Practice Questions Learning Spreadsheet: Exotic Options
Study Notes Learning Spreadsheet: Exotic Options
Chapter 16: Properties of Interest Rates
Study Notes: Properties of Interest Rates
Practice Question Set: Properties of Interest Rates
Instructional Video: Properties of Interest Rates
Learning Spreadsheet: Properties of Interest Rates
Chapter 17: Corporate Bonds
Study Notes: Corporate Bonds
Practice Question Set: Corporate Bonds
Instructional Video: Corporate Bonds
Chapter 18: Mortgages and Mortgage-Backed Securities
Study Notes: Mortgages and Mortgage-Backed Securities
Practice Question Set: Mortgages and Mortgage-Backed Securities
Instructional Video: Mortgages and Mortgage-Backed Securities
Learning Spreadsheet: Mortgages and Mortgage-Backed Securities
Chapter 19: Interest Rate Futures
Study Notes: Interest Rate Futures
Practice Question Set: Interest Rate Futures
Instructional Video: Interest Rate Futures
Learning Spreadsheet: Interest Rate Futures
Chapter 20: Swaps
Study Notes: Swaps
Practice Question Set: Swaps
Instructional Video: Swaps
Learning Spreadsheet: Swaps
Financial Markets & Products Topic Review
Financial Markets & Products Focus Review Video (1 of 3)
Financial Markets & Products Focus Review Video (2 of 3)
Financial Markets & Products Focus Review Video (3 of 3)
Learning Spreadsheets: P1.T3.a XLS Bundle
Learning Spreadsheets: P1.T3.b XLS Bundle
Learning Spreadsheets: P1.T3.c XLS Bundle
Learning Spreadsheets: P1.T3.d XLS Bundle
Valuation & Risk Models
Chapter 1: Measures of Financial Risk
Study Notes: Measures of Financial Risk
Practice Question Set: Measures of Financial Risk
Instructional Video: Measures of Financial Risk
Learning Spreadsheet: Measures of Financial Risk
Chapter 2: Calculating and Applying VaR
Study Notes: Calculating and Applying VaR
Practice Question Set: Calculating and Applying VaR
Instructional Video: Calculating and Applying VaR
Learning Spreadsheet: Calculating and Applying VaR
Chapter 3: Measuring and Monitoring Volatility
Study Notes: Measuring and Monitoring Volatility
Practice Question Set: Measuring and Monitoring Volatility
Instructional Video: Measuring and Monitoring Volatility
Learning Spreadsheet: Measuring and Monitoring Volatility
Chapter 4. External and Internal Ratings
Study Notes: External and Internal Ratings
Practice Question Set: External and Internal Ratings
Instructional Video: External and Internal Ratings
Learning Spreadsheet: External and Internal Ratings
Chapter 5: Country Risk: Country Risk: Determinants, Measures, and Implications
Study Notes: Country Risk: Country Risk: Determinants, Measures, and Implications
Practice Question Set: Country Risk: Country Risk: Determinants, Measures, and Implications
Instructional Video: Country Risk: Country Risk: Determinants, Measures, and Implications
Chapter 6: Measuring Credit Risk
Study Notes: Measuring Credit Risk
Practice Question Set: Measuring Credit Risk
Learning Spreadsheet: Measuring Credit Risk
Chapter 7: Operational Risk
Study Notes: Operational Risk
Practice Questions: Operational Risk
Learning Spreadsheet: Operational Risk
Chapter 8: Stress Testing
Study Notes: Stress Testing
Practice Question Set: Stress Testing
Instructional Video: Stress Testing
Chapter 9: Pricing Conventions, Discounting, and Arbitrage
Study Notes: Pricing Conventions, Discounting, and Arbitrage
Practice Question Set: Pricing Conventions, Discounting, and Arbitrage
Instructional Video: Pricing Conventions, Discounting, and Arbitrage
Learning Spreadsheet: Pricing Conventions, Discounting, and Arbitrage
Chapter 10: Interest Rates
Study Notes: Interest Rates
Practice Question Set: Interest Rates
Instructional Video: Interest Rates
Learning Spreadsheet: Interest Rates
Chapter 11: Bond Yields and Return Calculations
Study Notes: Bond Yields and Return Calculations
Practice Question Set: Bond Yields and Return Calculations
Instructional Video: Bond Yields and Return Calculations
Learning Spreadsheet: Bond Yields and Return Calculations
Chapter 12: Applying Duration, Convexity, and DV01
Study Notes: Applying Duration, Convexity, and DV01
Practice Question Set: Applying Duration, Convexity, and DV01
Instructional Video: Applying Duration, Convexity, and DV01
Learning Spreadsheet: Applying Duration, Convexity, and DV01
Chapter 13: Modeling and Hedging Non-Parallel Term Structure Shifts
Study Notes: Modeling and Hedging Non-Parallel Term Structure Shifts
Practice Question Set: Modeling and Hedging Non-Parallel Term Structure Shifts
Instructional Video: Modeling and Hedging Non-Parallel Term Structure Shifts
Learning Spreadsheet: Modeling and Hedging Non-Parallel Term Structure Shifts
Chapter 14: Binomial Trees
Study Notes: Binomial Trees
Practice Question Set: Binomial Trees
Instructional Video: Binomial Trees
Learning Spreadsheet: Binomial Trees
Chapter 15: The Black-Scholes-Merton Model
Study Notes: The Black-Scholes-Merton Model
Practice Question Set: The Black-Scholes-Merton Model
Instructional Video: The Black-Scholes-Merton Model
Learning Spreadsheet: The Black-Scholes-Merton Model
Chapter 16: Option Sensitivity Measures: The “Greeks”
Study Notes: Option Sensitivity Measures: The “Greeks”
Practice Question Set: Option Sensitivity Measures: The “Greeks”
Instructional Video: Option Sensitivity Measures: The “Greeks”
Learning Spreadsheet: Option Sensitivity Measures: The “Greeks”
Valuation & Risk Models Topic Review
Valuation and Risk Models Focus Review Video (1 of 3)
Valuation and Risk Models Focus Review Video (2 of 3)
Valuation and Risk Models Focus Review Video (3 of 3)
Learning Spreadsheets: P1.T4.a XLS Bundle
Learning Spreadsheets: P1.T4.b XLS Bundle
Learning Spreadsheets: P1.T4.c XLS Bundle
Learning Spreadsheets: P1.T4.e XLS Bundle
Expert-Led Live Online Review Sessions
Foundations of Risk Live Review – April 1, 2026 – 1:00p.m. Eastern Time
Foundations of Risk Pre-Session Video (Available March 27th)
Foundations of Risk Live Session Agenda & Access
Foundations of Risk Live Session Replay & Slides (available April 2nd)
Quantitative Analysis Live Review – April 8, 2026 – 1:00 p.m. Eastern Time
Quantitative Analysis Pre-Session Video
Quantitative Analysis Live Session Agenda & Access
Quantitative Analysis Live Session Replay & Slides (available April 9th)
Financial Markets & Products Live Review – April 15, 2026 – 1:00 p.m. Eastern Time
Financial Markets & Products Pre-Session Video (available April 10th)
Financial Markets & Products Live Session Agenda & Access
Financial Markets & Products Live Session Replay & Slides (available April 16th)
Valuation & Risk Models Live Review – April 22, 2026 – 1:00 p.m. Eastern Time
Valuation & Risk Models Pre-Session Video (available April 17th)
Valuation & Risk Models Live Session Agenda & Access
Valuation & Risk Models Live Session Replay & Slides (available April 23rd)
Part 1 Final Live Review – April 29, 2026 – 1:00 p.m. Eastern Time
Part 1 Final Review Pre-Session Video (available April 24th)
Part 1 Final Review Live Session Agenda & Access
Part 1 Final Review Live Session Replay & Slides (available April 30th)
Past Expert- Led Sessions
Foundations of Risk Live Review Sessions
Foundations of Risk Live Session Replay & Slides – May 2025 Exam
Foundations of Risk Live Session Replay & Slides – August 2025 Exam
Foundations of Risk Live Session Replay & Slides – November 2025 Exam
Quantitative Analysis Live Review Sessions
Quantitative Analysis Live Session Replay & Slides – May 2025 Exam
Quantitative Analysis Live Session Replay & Slides – August 2025 Exam
Quantitative Analysis Live Session Replay & Slides – November 2025
Financial Markets & Products Live Review Sessions
Financial Markets & Products Live Session Replay & Slides – May 2025 Exam
Financial Markets & Products Live Session Replay & Slides – August 2025 Exam
Financial Markets & Products Live Session Replay & Slides – November 2025 Exam
Valuation & Risk Models Live Review Sessions
Valuation & Risk Models Live Session Replay & Slides – May 2025 Exam
Valuation & Risk Models Live Session Replay & Slides – August 2025 Exam
Valuation & Risk Models Live Session Replay & Slides – November 2025 Exam
Part 1 Final Live Review Sessions
Part 1 Final Review Live Session Replay & Slides – May 2025 Exam
Final Live Review Live Session Replay & Slides – August 2025 Exam
Part 1 Final Review Live Session Replay & Slides – November 2025 Exam
PART I REVIEW
Part 1 Supplemental Mock Exams 1 & 2
Part 1 Mock Exam 1 (releases in January*)
Part 1 Mock Exam 2 (releases in February)
Formula Sheets
Part 1 Formula Sheets (Coming Soon)
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