Actuarial Notes

A First Course in Probability

By Sheldon Ross · Pearson · 2019

Study guide: Exam P-1

An elementary introduction to the theory of probability for students of mathematics, statistics, engineering and the sciences who know elementary calculus. Through numerous examples it presents both the mathematics of probability theory and its many possible applications. The tenth edition adds material on the Pareto distribution (Section 5.6.5), Poisson limit results (Section 8.5) and the Lorenz curve (Section 8.7), and new examples on NCAA basketball tournament win probabilities and the friendship paradox.

On the syllabus

  • Exam P — a suggested text (the syllabus requires none); Chapters 1, 2 and 3; Chapter 4 (exclude 4.8.4); Chapter 5 (exclude 5.6.2, 5.6.3, 5.6.5, 5.7); Chapter 6: 6.1, 6.2, 6.3.3, 6.3.4, 6.4, 6.6; Chapter 7 discrete only (exclude 7.2.1, 7.2.2, 7.3, 7.6, 7.7, 7.8, 7.9); Chapter 8: 8.1, 8.3.

Preface

1 Combinatorial Analysis

  • 1.1 Introduction
  • 1.2 The Basic Principle of Counting
  • 1.3 Permutations
  • 1.4 Combinations
  • 1.5 Multinomial Coefficients
  • 1.6 The Number of Integer Solutions of Equations

2 Axioms of Probability

  • 2.1 Introduction
  • 2.2 Sample Space and Events
  • 2.3 Axioms of Probability
  • 2.4 Some Simple Propositions
  • 2.5 Sample Spaces Having Equally Likely Outcomes
  • 2.6 Probability as a Continuous Set Function
  • 2.7 Probability as a Measure of Belief

3 Conditional Probability and Independence

4 Random Variables

5 Continuous Random Variables

6 Jointly Distributed Random Variables

  • 6.1 Joint Distribution Functions
  • 6.2 Independent Random Variables
  • 6.3 Sums of Independent Random Variables
    • 6.3.1 Identically Distributed Uniform Random Variables
    • 6.3.2 Gamma Random Variables
    • 6.3.3 Normal Random Variables
    • 6.3.4 Poisson and Binomial Random Variables
  • 6.4 Conditional Distributions: Discrete Case
  • 6.5 Conditional Distributions: Continuous Case
  • 6.6 Order Statistics
  • 6.7 Joint Probability Distribution of Functions of Random Variables
  • 6.8 Exchangeable Random Variables

7 Properties of Expectation

  • 7.1 Introduction
  • 7.2 Expectation of Sums of Random Variables
    • 7.2.1 Obtaining Bounds from Expectations via the Probabilistic Method
    • 7.2.2 The Maximum-Minimums Identity
  • 7.3 Moments of the Number of Events that Occur
  • 7.4 Covariance, Variance of Sums, and Correlations
  • 7.5 Conditional Expectation
    • 7.5.1 Definitions
    • 7.5.2 Computing Expectations by Conditioning
    • 7.5.3 Computing Probabilities by Conditioning
    • 7.5.4 Conditional Variance
  • 7.6 Conditional Expectation and Prediction
  • 7.7 Moment Generating Functions
    • 7.7.1 Joint Moment Generating Functions
  • 7.8 Additional Properties of Normal Random Variables
    • 7.8.1 The Multivariate Normal Distribution
    • 7.8.2 The Joint Distribution of the Sample Mean and Sample Variance
  • 7.9 General Definition of Expectation

8 Limit Theorems

  • 8.1 Introduction
  • 8.2 Chebyshev’s Inequality and the Weak Law of Large Numbers
  • 8.3 The Central Limit Theorem
  • 8.4 The Strong Law of Large Numbers
  • 8.5 Other Inequalities and a Poisson Limit Result
  • 8.6 Bounding the Error Probability When Approximating a Sum of Independent Bernoulli Random Variables by a Poisson Random Variable
  • 8.7 The Lorenz Curve

9 Additional Topics in Probability

10 Simulation

  • 10.1 Introduction
  • 10.2 General Techniques for Simulating Continuous Random Variables
    • 10.2.1 The Inverse Transformation Method
    • 10.2.2 The Rejection Method
  • 10.3 Simulating from Discrete Distributions
  • 10.4 Variance Reduction Techniques
    • 10.4.1 Use of Antithetic Variables
    • 10.4.2 Variance Reduction by Conditioning
    • 10.4.3 Control Variates

Answers to Selected Problems

Solutions to Self-Test Problems and Exercises

Index

Common Discrete Distributions

Common Continuous Distributions

Sources