The fundamental of Statistics

Probability theory is a branch of pure mathematics crucial for statistics field. In fact, probability theory provides a rigor mathematical description of statistical methods, where common statistical inference results such as frequentist properties of estimators, Classical and Bayesian methods have their fundamental in probability theory.

It was the Soviet mathematician Andrey Nikolaevich Kolmogorov in his work entitled “Foundations of the Theory of Probability (Berlin)” in 1933, who establishes the foundations of classical mathematical theory of probability as it is well known nowadays. Kolmogorov provided mathematical basis to represent random events by sets theory and probability is the measured defined on these sets.

In an advance probability course student learns the measure theory and the classical book Probability and Measure of the probability theorist and actor Patrick Billingsley is one primary reference.

In an intermediate level the measure theory is not addressed and no knowledge beyond elementary calculus is presumed. This restrictions of mathematical knowledge is necessary costly since more technical details are omitted, the generality of some theorems reduced, make statements without proof, and cumbersome arguments is sometimes have to be used. All these sacrifices, however, is inhibit the importance of the course rather less than one might suppose. The essential aspects of theory is entirely comprehensible without higher mathematics.

Course contents

In my Master’s Degree I had an intermediate course of probability theory. The textbook was “Probability: An Intermediate Level Course” by Barry Ree James (in memorial). The course was taught by professor Diego Fernando de Bernardini, who literally copied the book on board. During the course I organized some notes with the course contents and can be find here (in Portuguese).

The course was organized in the following major themes:

• Basic definitions

• Random variable and distribution functions

• Mathematical expectation

• Conditional distribution and expectation

• Law of large numbers

• Convergence in distribution

• Central Limit Theorem

Exams

At the end of the course I spent one month studying all the contents and solving a lot of exercises in order to prepare myself for the probability exam. I remembered to have solved all exercises of the previous exams and from Barry James book. My probability exam can be find here (in Portuguese).