PROBABILITY THEORY

The course is one of those characterizing Economia e Finanza, Curriculum ECONOMICS - QEM
The first part of the course is intended to be an introduction to fundamentals of probability theory in order to devote the second part to statistical inference. The use of these tools allows the analysis of macroeconomic as well as financial data and therefore the interpretation of a variety of economic phenomena.

1. KNOWLEDGE AND UNDERSTANDING
1.1 Understand probability theory fundamentals
1.2 Understand probability calculus via random variables
1.3 Understand statistical inference tools
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 Solve basic probability calculus problems
2.2 Solve probability problems with random variables
2.3 Solve statistical inference problems with estimators and hypothesis testing
3. MAKING JUDGMENTS
3.1 Being able to recongnize the correct theoretical tool to solve the problem
3.2 Being able to interpret the obtained results

statistics (basics of descriptive statistics and probability), mathematics

First Part:
Set definition, elementary operations with sets, basics of probability theory
Random variables, distribution functions, density and mass functions
Expected values, moments,
Common families of distributions,
Bivariate random variables, conditional distribution and independence, covariance and correlation
Random vectors

Second part:
Properties of a random sample: basic concepts, sums of random variables from a random sample, convergence concepts (CB chapter 5, sections 5.1, 5.2, 5.5)
Data reduction: sufficiency principle, likelihood principle, equivariance principle (CB chapter 6, sections 6.1,6.2.1, 6.3,6.4)
Point estimation: methods of finding estimators (method of moments, maximum likelihood), evaluating estimators (CB chapter 7, sections 7.1,7.2.1,7.2.2, 7.3)
Hypothesis testing: the likelihood ratio tests (LRTs), error probabilities and power function (CB chapter 8, sections 8.1,8.2.1,8.3.1)
Asymptotic evaluation: point estimation (consistency and efficiency), hypothesis testing (asymptotic distribution of LRTs) (CB chapter 10, sections 10.1.1,10.1.2,10.3.1)

Lecture notes, slides and exercises for the entire duration of course are made available using the Moodle pages of the course.
There are some suggested textbooks:
For Probability:
- Mood, A. et. al. (1974) Introduction to the Theory of Statistics, McGraw-Hill, Inc., NY (Chapters 1 to 5)
- Rice, J. (2007) Mathematical Statistics and Data Analysis, Thomson, Berkley, CA (Chapters 1, 2, 3, 4, 6)
For Inference:
- Casella, G. and Berger, R.L. (1990, 2002). Statistical Inference. Wadsworth publishing Co., Belmont, CA (Chapters 5 to 8)

Mid-term written exam after the first 5 weeks of course. Second written exam after the second 5 weeks of course. The final grade will be given by the average of the two previous grades.
In the following exam sessions from January to September and the written exam will be a single comprehensive examination

The programme will develop with a careful balance of teaching and learning. This is delivered by Lectures where theorethical concepts are presented alternated with exercise sessions that are solved in class.

English

written