Actuarial Notes

Probability for Risk Management

By Matthew J. Hassett, Donald G. Stewart and Jelena Milovanovic · ACTEX Learning · 2021

Study guide: Exam P-1

A first course in probability for students with a basic calculus background who are mostly interested in its applications to risk management. The authors name insurance, finance, economics and the health sciences as those areas, claim complete coverage of the SOA Exam P syllabus, and favour a practical, intuitive style over formal proofs. The third edition revises the text throughout, enhancing or adding the bivariate normal distribution, probability generating functions, the coefficient of variation, order statistics and the correlation coefficient.

On the syllabus

  • Exam P — a suggested text (the syllabus requires none); Chapters 1, 2, 3, 4 and 5; Chapter 6: 6.1, 6.2.1; Chapter 7; Chapter 8 (exclude 8.5, 8.6, 8.7); Chapter 9 (exclude 9.2, 9.3, 9.4, 9.6); Chapter 10 (exclude 10.2, 10.3.2, 10.3.3 continuous, 10.4.2); Chapter 11 (exclude 11.1.4, 11.1.5, 11.2.3 continuous, 11.2.5 continuous, 11.2.8, 11.3).

Preface to the Third Edition

1 Probability: A Tool for Risk Management

  • 1.1 Who Uses Probability?
  • 1.2 An Example from Insurance
  • 1.3 Probability and Statistics
  • 1.4 Some History
  • 1.5 Computing Technology

2 Counting for Probability

  • 2.1 What is Probability?
  • 2.2 The Language of Probability: Sets, Sample Spaces, and Events
  • 2.3 Compound Events; Set Notation
    • 2.3.1 Negation
    • 2.3.2 The Compound Events E or F, E and F
    • 2.3.3 New Sample Spaces from Old; Ordered Pair Outcomes
  • 2.4 Set Identities
    • 2.4.1 The Distributive Laws for Sets
    • 2.4.2 De Morgan’s Laws
  • 2.5 Counting
    • 2.5.1 Basic Rules
    • 2.5.2 Using Venn Diagrams in Counting Problems
    • 2.5.3 Trees
    • 2.5.4 The Multiplication Principle for Counting
    • 2.5.5 Permutations
    • 2.5.6 Combinations
    • 2.5.7 Combined Problems
    • 2.5.8 Partitions
    • 2.5.9 Some Useful Identities
  • 2.6 Exercises
  • 2.7 Sample Actuarial Examination Problems

3 Elements of Probability

  • 3.1 Probability by Counting for Equally Likely Outcomes
    • 3.1.1 Definition of Probability for Equally Likely Outcomes
    • 3.1.2 Probability Rules for Compound Events
    • 3.1.3 More Counting Problems
  • 3.2 Probability When Outcomes Are Not Equally Likely
    • 3.2.1 Assigning Probabilities to a Finite Sample Space
    • 3.2.2 The General Definition of Probability
  • 3.3 Conditional Probability
    • 3.3.1 Conditional Probability by Counting
    • 3.3.2 Defining Conditional Probability
    • 3.3.3 Using Trees in Probability Problems
    • 3.3.4 Conditional Probabilities in Life Tables
  • 3.4 Independence
    • 3.4.1 An Example of Independent Events; The Definition of Independence
    • 3.4.2 The Multiplication Rules for Independent Events
  • 3.5 Bayes’ Theorem
  • 3.6 Exercises
  • 3.7 Sample Actuarial Examination Problems

4 Discrete Random Variables

  • 4.1 Random Variables
    • 4.1.1 Definition of Probability for Equally Likely Outcomes
    • 4.1.2 Redefining a Random Variable
    • 4.1.3 Notation; The Distinction Between X and x
  • 4.2 The Probability Function of a Discrete Random Variable
  • 4.3 Measuring Central Tendency: Expected Value
    • 4.3.1 Central Tendency: The Mean
    • 4.3.2 The Expected Value of Y = aX
    • 4.3.3 The Median
    • 4.3.4 The Mode
  • 4.4 Variance and Standard Deviation
    • 4.4.1 Measuring Variation
    • 4.4.2 The Variance and Standard Deviation of Y = aX
    • 4.4.3 Comparing Two Stocks
    • 4.4.4 Coefficient of Variation
    • 4.4.5 z-scores; Chebyshev’s Theorem
  • 4.5 Population and Sample Statistics
    • 4.5.1 Population and Sample Mean
    • 4.5.2 Using Calculators for the Mean and Standard Deviation
  • 4.6 Dealing with More Than One Random Variable
  • 4.7 Exercises
  • 4.8 Sample Actuarial Exam Problems

5 Commonly Used Discrete Distributions

  • 5.1 The Binomial Distribution
    • 5.1.1 Binomial Random Variables
    • 5.1.2 Binomial Probabilities
    • 5.1.3 Mean and Variance of the Binomial Distribution
    • 5.1.4 Applications
    • 5.1.5 Checking Assumptions for Binomial Problems
  • 5.2 The Hypergeometric Distribution
    • 5.2.1 An Example
    • 5.2.2 The Hypergeometric Distribution
    • 5.2.3 The Mean and Variance of the Hypergeometric Distribution
    • 5.2.4 Relating the Binomial and Hypergeometric Distributions
  • 5.3 The Poisson Distribution
    • 5.3.1 The Poisson Distribution
    • 5.3.2 The Poisson Approximation to the Binomial for Large n and Small p
    • 5.3.3 Why Poisson Probabilities Approximate Binomial Probabilities
    • 5.3.4 Derivation of the Expected Value of a Poisson Random Variable
  • 5.4 The Geometric Distribution
    • 5.4.1 Waiting Time Problems
    • 5.4.2 The Mean and Variance of the Geometric Distribution
    • 5.4.3 The Cumulative Distribution for the Geometric Distribution
    • 5.4.4 An Alternate Formulation of the Geometric Distribution
    • 5.4.5 Using the Survival Function to Find the Mean of a Counting Random Variable
    • 5.4.6 The Memoryless Property of the Geometric Distribution
  • 5.5 The Negative Binomial Distribution
    • 5.5.1 Relation to the Geometric Distribution
    • 5.5.2 The Mean and Variance of the Negative Binomial Distribution
  • 5.6 The Discrete Uniform Distribution
  • 5.7 Exercises
  • 5.8 Sample Actuarial Examination Problems

6 Applications for Discrete Random Variables

  • 6.1 Functions of Random Variables and Their Expectations
    • 6.1.1 The Function Y = aX + b
    • 6.1.2 Analyzing Y=f(X) in General
    • 6.1.3 Expected Value of a Loss or Claim
    • 6.1.4 Expected Utility
  • 6.2 Moments and the Moment Generating Function
    • 6.2.1 Moments of a Random Variable
    • 6.2.2 The Moment Generating Function
    • 6.2.3 Moment Generating Function for the Binomial Random Variable
    • 6.2.4 Moment Generating Function for the Poisson Random Variable
    • 6.2.5 Moment Generating Function for the Geometric Random Variable
    • 6.2.6 Moment Generating Function for the Negative Binomial Random Variable
    • 6.2.7 Moment Generating Function for the Discrete Uniform Random Variable
    • 6.2.8 Other Uses of the Moment Generating Function
    • 6.2.9 Useful Identity and Shortcuts
    • 6.2.10 Infinite Series and the Moment Generating Function
    • 6.2.11 The Probability Generating Function
  • 6.3 Distribution Shapes
  • 6.4 Simulation of Discrete Distributions
    • 6.4.1 A Coin-Tossing Example
    • 6.4.2 Generating Random Numbers from [0,1)
    • 6.4.3 Simulating Any Finite Discrete Distribution
    • 6.4.4 Simulating A Binomial Distribution
    • 6.4.5 Simulating A Geometric Distribution
    • 6.4.6 Simulating A Negative Binomial Distribution
    • 6.4.7 Simulating Other Distributions
  • 6.5 Exercises
  • 6.6 Sample Actuarial Exam Problems

7 Continuous Random Variables

  • 7.1 Defining a Continuous Random Variable
    • 7.1.1 A Basic Example
    • 7.1.2 The Density Function and Probabilities for Continuous Random Variables
    • 7.1.3 Building a Straight-Line Density Function for an Insurance Loss
    • 7.1.4 The Cumulative Distribution Function F(x)
    • 7.1.5 A Piecewise Density Function
  • 7.2 The Mode, the Median, and Percentiles
  • 7.3 The Mean and Variance of a Continuous Random Variable
    • 7.3.1 The Expected Value of a Continuous Random Variable
    • 7.3.2 The Expected Value of a Function of a Random Variable
    • 7.3.3 Finding E(X) using F(X)
    • 7.3.4 The Variance of a Continuous Random Variable
  • 7.4 Dealing with More Than One Random Variable
  • 7.5 Exercises
  • 7.6 Sample Actuarial Examination Problems

8 Commonly Used Continuous Distribution

  • 8.1 The Uniform Distributions
    • 8.1.1 The Uniform Density Function
    • 8.1.2 The Cumulative Distribution Function for a Uniform Random Variable
    • 8.1.3 Uniform Random Variables for Lifetimes; Survival Functions
    • 8.1.4 The Mean and Variance of the Uniform Distribution
    • 8.1.5 A Conditional Probability Problem Involving the Uniform Distribution
  • 8.2 The Exponential Distribution
    • 8.2.1 Mathematical Preliminaries
    • 8.2.2 The Exponential Density: An Example
    • 8.2.3 The Exponential Density Function
    • 8.2.4 The Cumulative Distribution Function and Survival Function of the Exponential Random Variable
    • 8.2.5 The Mean and Variance of the Exponential Distribution
    • 8.2.6 Another Look at the Meaning of the Density Function
    • 8.2.7 The Failure (Hazard) Rate
    • 8.2.8 Use of the Cumulative Distribution Function
    • 8.2.9 Why the Waiting Time Is Exponential for Events Whose Number Follows a Poisson Distribution
    • 8.2.10 A Conditional Probability Problem Involving the Exponential Distribution
  • 8.3 The Gamma Distribution
    • 8.3.1 Applications of the Gamma Distribution
    • 8.3.2 The Gamma Density Function
    • 8.3.3 Sums of Independent Exponential Random Variables
    • 8.3.4 The Mean and Variance of the Gamma Distribution
    • 8.3.5 Notational Differences Between Texts
  • 8.4 The Normal Distribution
    • 8.4.1 Applications of the Normal Distribution
    • 8.4.2 The Normal Density Function
    • 8.4.3 Calculation of Normal Probabilities; The Standard Normal
    • 8.4.4 Sums of Independent, Identically Distributed, Random Variables
    • 8.4.5 Percentiles of the Normal Distribution
    • 8.4.6 The Continuity Correction
  • 8.5 The Lognormal Distribution
    • 8.5.1 Applications of the Lognormal Distribution
    • 8.5.2 Defining the Lognormal Distribution
    • 8.5.3 Calculating Probabilities for a Lognormal Random Variable
    • 8.5.4 The Lognormal Distribution for a Stock Price
  • 8.6 The Pareto Distribution
    • 8.6.1 Application of the Pareto Distribution
    • 8.6.2 The Density Function of the Pareto Random Variable
    • 8.6.3 The Cumulative Distribution Function; Evaluating Probabilities
    • 8.6.4 The Mean and Variance of the Pareto Distribution
    • 8.6.5 The Failure Rate of a Pareto Random Variable
  • 8.7 The Weibull Dstribution
    • 8.7.1 Application of the Weibull Distribution
    • 8.7.2 The Density Function of the Weibull Distribution
    • 8.7.3 The Cumulative Distribution Function and Probability Calculations
    • 8.7.4 The Mean and Variance of the Weibull Distribution
    • 8.7.5 The Failure Rate of a Weibull Random Variable
  • 8.8 The Beta Distribution
    • 8.8.1 Applications of the Beta Distribution
    • 8.8.2 The Density Function of the Beta Distribution
    • 8.8.3 The Cumulative Distribution Function and Probability Calculations
    • 8.8.4 A Useful Identity
    • 8.8.5 The Mean and Variance of the Beta Random Variable
  • 8.9 Fitting Theoretical Distributions to Real Problems
  • 8.10 Exercises
  • 8.11 Sample Actuarial Problems

9 Applications for Continuous Random Variables

  • 9.1 Expected Value of a Function of a Random Variable
    • 9.1.1 Calculating E(g(X))
    • 9.1.2 Expected Value of a Loss or Claim
    • 9.1.3 Expected Utility
  • 9.2 Moment Generating Functions for Continuous Random Variables
    • 9.2.1 A Review
    • 9.2.2 The Gamma Moment Generating Function
    • 9.2.3 The Normal Moment Generating Function
  • 9.3 The Distribution of Y=g(X)
    • 9.3.1 An Example
    • 9.3.2 Using F_X(x) to Find F_Y(y) for Y = g(X)
    • 9.3.3 Finding the Density Function for Y=g(X) When g(X) Has an Inverse Function
  • 9.4 Simulation of Continuous Distributions
    • 9.4.1 The Inverse Cumulative Distribution Function Method
    • 9.4.2 Using the Inverse Transformation Method to Simulate an Exponential Random Variable
    • 9.4.3 Simulating Other Distributions
  • 9.5 Mixed Distributions
    • 9.5.1 An Insurance Example
    • 9.5.2 The Probability Function for a Mixed Distribution
    • 9.5.3 The Expected value of a Mixed Distribution
    • 9.5.4 A Lifetime Example
  • 9.6 A Useful Identity
    • 9.6.1 Using the Hazard Rate to Find the Survival Function
  • 9.7 Exercises
  • 9.8 Sample Actuarial Examination Problems

10 Multivariate Distributions

  • 10.1 Joint Distributions for Discrete Random Variables
  • 10.2 Joint Distributions for Continuous Random
    • 10.2.1 Review of the Single Variable Case
    • 10.2.2 The Joint Probability Density Function for Two Continuous Random Variables
    • 10.2.3 Marginal Distributions for Continuous Random Variables
    • 10.2.4 Using Continuous Marginal Distributions
    • 10.2.5 More General Joint Probability Calculations
  • 10.3 Conditional Distributions
  • 10.4 Independence for Random Variables
    • 10.4.1 Independence for Discrete Random Variables
    • 10.4.2 Independence for Continuous Random Variables
  • 10.5 The Multinomial Distribution
  • 10.6 Exercises
  • 10.7 Sample Actuarial Examination Problems

11 Applying Multivariate Distributions

  • 11.1 Distributions of Functions of Two Random Variables
    • 11.1.1 Functions of X and Y
    • 11.1.2 The Sum of Two Discrete Random Variables
    • 11.1.3 The Sum of Independent Discrete Random Variables
    • 11.1.4 The Sum of Continuous Random Variables
    • 11.1.5 The Sum of Independent Continuous Random Variables
    • 11.1.6 The Minimum of Two Independent Exponential Random Variables
    • 11.1.7 The Minimum and Maximum of any Two Independent Random Variables
    • 11.1.8 Order Statistics
  • 11.2 Expected Values of Functions of Random Variables
    • 11.2.1 Finding E(g(X,Y))
    • 11.2.2 Finding E(X+Y)
    • 11.2.3 The Expected Value of XY
    • 11.2.4 The Covariance of X and Y
    • 11.2.5 The Variance of X+Y
    • 11.2.6 Useful Properties of Covariance
    • 11.2.7 The Correlation Coefficient
    • 11.2.8 The Bivariate Normal Distribution
  • 11.3 Moment Generating Functions for Sums of Independent Random Variables; Joint Moment Generating Functions
    • 11.3.1 The General Principle
    • 11.3.2 The Sum of Independent Poisson Random Variables
    • 11.3.3 The Sum of Independent and Identically Distributed Geometric Random Variables
    • 11.3.4 The Sum of Independent Normal Random Variables
    • 11.3.5 The Sum of Independent and Identically Distributed Exponential Random Variables
    • 11.3.6 Joint Moment Generating Functions
  • 11.4 The Sum of More Than Two Random Variables
    • 11.4.1 Extending the Results of Section 11.3
    • 11.4.2 The Mean and Variance of X + Y + Z
    • 11.4.3 The Sum of a Large Number of Independent and Identically Distributed Random Variables
  • 11.5 Double Expectation Theorems
    • 11.5.1 Conditional Expectations
    • 11.5.2 Conditional Variances
  • 11.6 Applying the Double Expectation Theorem; The Compound Poisson Distribution
    • 11.6.1 The Total Claim Amount for an Insurance Company: An Example of the Compound Poisson Distribution
    • 11.6.2 The Mean and Variance of a Compound Poisson Random Variable
    • 11.6.3 Derivation of the Mean and Variance Formulas
    • 11.6.4 Finding Probabilities for the Compound Poisson S by a Normal Approximation
  • 11.7 Exercises
  • 11.8 Sample Actuarial Examination Problems

12 Stochastic Processes

  • 12.1 Simulation Examples
    • 12.1.1 Gambler’s Ruin Problem
    • 12.1.2 Fund Switching
    • 12.1.3 A Compound Poisson Process
    • 12.1.4 A Continuous Process: Simulating Exponential Waiting Times
    • 12.1.5 Simulation and Theory
  • 12.2 Finite Markov Chains
    • 12.2.1 Examples
    • 12.2.2 Probability Calculations for Markov Processes
  • 12.3 Regular Markov Processes
    • 12.3.1 Basic Properties
    • 12.3.2 Finding the Limiting Matrix of a Regular Finite Markov Chain
  • 12.4 Absorbing Markov Chains
    • 12.4.1 Another Gambler’s Ruin Example
    • 12.4.2 Probabilities of Absorption
  • 12.5 Further Study of Stochastic Processes
  • 12.6 Exercises

Appendix A

Appendix B

Answers to the Exercises

Bibliography

Sources