MATH 471 Syllabus

Problem Solving for Actuarial Exam P

Revised: September 2014

Course Description

Advanced problem solving by means of extensive review and practice. Preparing students for Exam P of the Society of Actuaries and the Casualty Actuarial Society. MATH 370. Three semester hours.

Student Learning Objectives

From the “learning objectives” listed for this professional exam, candidates (students) should be able to use and apply the following concepts in a risk management context:

• General Probability (Set functions including set notation and basic elements of probability, Mutually exclusive events, Addition and multiplication rules, Independence of events, Combinatorial probability, Conditional probability, Bayes Theorem/Law of total probability);
• Univariate probability distributions (including binomial, negative binomial, geometric, hypergeometric, Poisson, uniform, exponential, gamma, and normal), Probability functions and probability density functions, Cumulative distribution functions, Mode, median, percentiles, and moments, Variance and measures of dispersion, Moment generating functions, Transformations; abd
• Multivariate probability distributions (including the bivariate normal), Joint probability functions and joint probability density functions, Joint cumulative distribution functions, Central Limit Theorem, Conditional and marginal probability distributions, Moments for joint, conditional, and marginal probability distributions, Joint moment generating functions, Variance and measures of dispersion for conditional and marginal probability distributions, Covariance and correlation coefficients, Transformations and order statistics, Probabilities and moments for linear combinations of independent random variables.

Text

Wackerley, Dennis, Mendenhall, William., and Scheaffer, Richard. Introduction to Mathematical Statistics with Applications, Seventh Edition. Brooks/Cole Cengage, 2008.

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• Chapter 2: Probability (7 days) Introduction. Probability and Inference. A Review of Set Notation. A Probabilistic Model for an Experiment: The Sample-Point Method. Tools for Counting Sample Points. Conditional Probability and the Independence of Events. Two Laws of Probability. Calculating the Probability of an Event: The Event-Composition Method. The Law of Total Probability and Bayes’ Rule. Numerical Events and Random Variables. Random Sampling. Summary.
• Chapter 3: Discrete Random Variables and Their Probability Distributions (7 days) Basic Definition. The Probability Distribution for a Discrete Random Variable. The Expected Value of a Random Variable or a Function of a Random Variable. The Binomial Probability Distribution. The Geometric Probability Distribution. The Negative Binomial Probability Distribution. The Hypergeometric Probability Distribution. The Poisson Probability Distribution. Moment and Moment-Generating Functions. Probability-Generating Functions. Tchebysheff's Theorem. Summary.
• Chapter 4: Continuous Variables and Their Probability Distributions (7 days) Introduction. The Probability Distribution for a Continuous Random Variable. Expected Values for Continuous Random Variables. The Uniform Probability Distribution. The Normal Probability Distribution. The Gamma Probability Distribution. The Beta Probability Distribution. Some General Comments. Other Expected Values. Tchebysheff's Theorem. Expectations of Discontinuous Functions and Mixed Probability Distributions. Summary.
• Chapter 5: Multivariate Probability Distributions (10 days) Introduction. Bivariate and Multivariate Probability Distributions. Marginal and Conditional Probability Distributions. Independent Random Variables. The Expected Value of a Function of Random Variables. Special Theorems. The Covariance of Two Random Variables. The Expected Value and Variance of Linear Function of Random Variables. The Multinomial Probability Distribution. The Bivariate Normal Distribution. Conditional Expectation. Summary.
• Chapter 6: Functions of Random Variables (5 days) Introduction. Finding the Probability Function of a Function of Random Variables. The Method of Distribution Functions. The Method of Transformations. The Method of Moment-Generating Functions. Multivariable Transformations Using Jacobians. Order Statistics. Summary.
• Chapter 7: Sampling Distributions and the Central Limit Theorem (4 days) Introduction. Sampling Distributions Related to the Normal Distribution. The Central Limit Theorem. A Proof of the Central Limit Theorem (Optional). The Normal Approximation to the Binomial Distribution. Summary.