![[Probability-for-Risk-Management-Hassett-2021-Cover.jpg]] ## 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 Definition of Independence - 3.4.2 Multiplication Rules for Independent Events - 3.5 Bayes’ Theorem - 3.5.1 Testing a Test: An Example - 3.5.2 Law of Total Probability and Bayes’ Theorem - 3.6 Exercises - 3.7 Sample Actuarial Examination Problems ## 4 Discrete Random Variables - 4.1 Random Variables - 4.1.1 Definition for Equally Likely Outcomes - 4.1.2 Redefining a Random Variable - 4.1.3 Notation: X vs x - 4.2 Probability Function - 4.2.1 Defining the Probability Function - 4.2.2 The Cumulative Distribution Function - 4.3 Expected Value - 4.3.1 Mean - 4.3.2 Expected Value of aX - 4.3.3 Median - 4.3.4 Mode - 4.4 Variance and Standard Deviation - 4.4.1 Measuring Variation - 4.4.2 Variance of 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 Means - 4.5.2 Calculator Methods - 4.6 Multiple Random Variables - 4.6.1 Expected Value of X + Y - 4.6.2 Independence - 4.6.3 Variance of X + Y - 4.7 Exercises - 4.8 Sample Actuarial Exam Problems ## 5 Commonly Used Discrete Distributions - 5.1 Binomial Distribution - 5.1.1 Definition - 5.1.2 Probabilities - 5.1.3 Mean and Variance - 5.1.4 Applications - 5.1.5 Assumptions - 5.2 Hypergeometric Distribution - 5.2.1 Example - 5.2.2 Definition - 5.2.3 Mean and Variance - 5.2.4 Relation to Binomial - 5.3 Poisson Distribution - 5.3.1 Definition - 5.3.2 Approximation to Binomial - 5.3.3 Why Approximation Works - 5.3.4 Expected Value - 5.4 Geometric Distribution - 5.4.1 Waiting Time - 5.4.2 Mean and Variance - 5.4.3 CDF - 5.4.4 Alternate Form - 5.4.5 Survival Function Method - 5.4.6 Memoryless Property - 5.5 Negative Binomial Distribution - 5.5.1 Relation to Geometric - 5.5.2 Mean and Variance - 5.6 Discrete Uniform Distribution - 5.7 Exercises - 5.8 Sample Actuarial Examination Problems ## 6 Applications for Discrete Random Variables - 6.1 Functions of Random Variables - 6.1.1 Linear Functions - 6.1.2 General Functions - 6.1.3 Expected Loss - 6.1.4 Expected Utility - 6.2 Moments and Moment Generating Functions - 6.2.1 Moments - 6.2.2 MGF Definition - 6.2.3–6.2.7 MGFs for Common Distributions - 6.2.8 Other Uses - 6.2.9 Identities and Shortcuts - 6.2.10 Infinite Series - 6.2.11 Probability Generating Function - 6.3 Distribution Shapes - 6.4 Simulation of Discrete Distributions - 6.4.1–6.4.7 Simulation Methods - 6.5 Exercises - 6.6 Sample Actuarial Exam Problems ## 7 Continuous Random Variables - 7.1 Definition and Density Functions - 7.1.1 Basic Example - 7.1.2 Density and Probabilities - 7.1.3 Insurance Example - 7.1.4 CDF - 7.1.5 Piecewise Densities - 7.2 Mode, Median, Percentiles - 7.3 Mean and Variance - 7.4 Multiple Variables - 7.5 Exercises - 7.6 Sample Actuarial Examination Problems ## 8 Commonly Used Continuous Distributions - 8.1 Uniform Distribution - 8.2 Exponential Distribution - 8.3 Gamma Distribution - 8.4 Normal Distribution - 8.5 Lognormal Distribution - 8.6 Pareto Distribution - 8.7 Weibull Distribution - 8.8 Beta Distribution - 8.9 Fitting Distributions - 8.10 Exercises - 8.11 Sample Actuarial Problems ## 9 Applications for Continuous Random Variables - 9.1 Expected Value of Functions - 9.2 Moment Generating Functions - 9.3 Distribution of Y = g(X) - 9.4 Simulation Methods - 9.5 Mixed Distributions - 9.6 Useful Identity (Hazard Rate) - 9.7 Exercises - 9.8 Sample Actuarial Examination Problems ## 10 Multivariate Distributions - 10.1 Joint Distributions (Discrete) - 10.2 Joint Distributions (Continuous) - 10.3 Conditional Distributions - 10.4 Independence - 10.5 Multinomial Distribution - 10.6 Exercises - 10.7 Sample Actuarial Examination Problems ## 11 Applying Multivariate Distributions - 11.1 Functions of Two Random Variables - 11.2 Expected Values and Covariance - 11.3 Moment Generating Functions (Joint) - 11.4 Sums of Multiple Variables - 11.5 Double Expectation Theorems - 11.6 Compound Poisson Distribution - 11.7 Exercises - 11.8 Sample Actuarial Examination Problems ## 12 Stochastic Processes - 12.1 Simulation Examples - 12.2 Finite Markov Chains - 12.3 Regular Markov Processes - 12.4 Absorbing Markov Chains - 12.5 Further Study - 12.6 Exercises