# PrEMAS 3 - Probability theory and statistics for actuaries

## The module aims at introducing students to fundamental notions of probability theory and statistics. Probability and statistics are important practical tools in the actuary’s daily work.

This module covers their principles, such as: sample spaces, probability, independence, distributions, expectation and conditional expectation, moment generating functions, limit theorems, point estimation, resampling methods, confidence intervals, hypothesis testing, analysis of variance, Bayesian inference, and others. The student will get hands-on experience with solution of a number of standard probabilistic and statistical tasks. Relevant methods will be illustrated using the statistical software package R.

### Entry requirements

It is strongly advised that students have finished the module PrEMAS 1: Mathematical Methods for Actuaries (or equivalent) before starting this course.

### Learning objectives

The module consists of two parts. In the probability theory part, the student

• will be able to define probability spaces, events and probability;
• will learn the notions of independence and conditional probability;
• will be able to perform elementary computations with probabilities and conditional probabilities;
• will learn the concepts of a random variable, distribution, density and quantile functions, and will be able to compute moments of a number of standard distributions;
• will learn the notion of independence of random variables and various properties of conditional expectation;
• will learn the concept of moment generating functions, will be able to compute them for a number of standard probability distributions, and will learn their applications and uses;
• will learn a number of classical probability inequalities, like Chebyshev and Hoeffding, and their applications;
• will learn convergence notions used in probability theory and several classical limit theorems, and will be able to apply them in various contexts.

In the statistics part, the student

• will learn about statistical models, and fundamental issues addressed by statistical inference (estimation, testing, uncertainty quantification);
• will learn the notion of the empirical distribution function and its applications in estimation of statistical functionals;
• will learn about and will be able to apply resampling methods (bootstrap and jackknife) to various statistical problems;
• will learn about fundamentals of parametric inference, classical inference techniques like the method of moments and the maximum likelihood, the delta method, and will be able to carry out estimation tasks in a range of statistical models;
• will learn about several classical statistical tests and will be able to apply them in various settings;
• will learn about and will be able to apply the principles of linear regression analysis;
• will learn about principles of the Bayesian inference and perform Bayesian analysis for a range of statistical models.

### Literature

Compulsory literature

• Larry Wasserman, All of Statistics: A Concise Course in Statistical Inference - ISBN 978-0-387-40272-7. Other versions of this book can also be used.