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)

In each academic year there are 4 exam sessions. The first session takes place during the first semester (while the course is being taught), and it is articulated in a midterm partial written examination administered at the end of the first teaching period and a final partial written examination administered at the end of the second teaching period. The final evaluation is obtained as average of the two partial examinations. The remaining sessions (from January until September) present a single comprehensive examination on the full contents of the course.

The lecturer has a duty to ensure that the rules regarding the authenticity and originality of exam tests and papers are respected. Therefore, if there is suspicion of irregular conduct, an additional assessment may be conducted, which could differ from the original exam description.

written

The lecturer has a duty to ensure that the rules regarding the authenticity and originality of exam tests and papers are respected. Therefore, if there is suspicion of irregular conduct, an additional assessment may be conducted, which could differ from the original exam description.

Regarding the grading scale (criteria for assigning grades):
A. Scores in the 18-22 range will be assigned in the presence of:
Sufficient knowledge and understanding of the course program;
Limited ability to apply knowledge and formulate independent judgments;
Sufficient ability to communicate using the appropriate technical language of the subject.

B. Scores in the 23-26 range will be assigned in the presence of:
Fair knowledge and understanding of the course program;
Fair ability to apply knowledge and formulate independent judgments;
Fair ability to communicate using the appropriate technical language of the subject.

C. Scores in the 27-30 range will be assigned in the presence of:
Good to excellent knowledge and understanding of the course program;
Good to excellent ability to apply knowledge and formulate independent judgments;
Good to excellent ability to communicate using the appropriate technical language of the subject.
D. Honors will be awarded in the presence of outstanding knowledge and applied understanding of the program, excellent judgment skills, and exceptional communication abilities.

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.