## Table of Contents - **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 and Example - 3.4.2 Multiplication Rules for Independent Events - 3.5 Bayes’ Theorem - 3.5.1 Testing a Test: Example - 3.5.2 Law of Total Probability; Bayes’ Theorem - 3.6 Exercises - 3.7 Sample Actuarial Examination Problems - **4. Discrete Random Variables** - 4.1 Random Variables - 4.1.1 Definition (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 Cumulative Distribution Function - 4.3 Expected Value - 4.3.1 Mean - 4.3.2 Expected Value of Y = 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 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 - 4.6 Multiple Random Variables - 4.7 Exercises - 4.8 Sample Actuarial Exam Problems - **5. Commonly Used Discrete Distributions** - 5.1 Binomial Distribution - 5.2 Hypergeometric Distribution - 5.3 Poisson Distribution - 5.4 Geometric Distribution - 5.5 Negative Binomial Distribution - 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 - 6.2 Moment Generating Function - 6.3 Distribution Shapes - 6.4 Simulation - 6.5 Exercises - 6.6 Sample Actuarial Exam Problems - **7. Continuous Random Variables** - 7.1 Defining a Continuous Random Variable - 7.2 Mode, Median, Percentiles - 7.3 Mean and Variance - 7.4 Multiple Random 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 - 9.5 Mixed Distributions - 9.6 Useful Identity - 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 - 11.3 Moment Generating Functions - 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