29596 - Probability And Statistics

Learning outcomes

Upon successful completion of the 30‑hour module, students will be able to:

1. Formulate rigorous probability models
   Construct an appropriate sample space, σ‑algebra and probability measure for economic phenomena.

2. Translate economic questions into probabilistic statements
   Deploy counting arguments, conditional probability and Bayes’ theorem to evaluate event probabilities in settings such as lotteries, information games and urn models.

3. Analyse random variables and their distributions
   Identify relevant discrete or continuous laws, derive their moments and transforms, and describe joint, marginal and conditional behaviour.

4. Apply large‑sample theory
   State and use the main modes of convergence, justify the Weak/Strong Law of Large Numbers and Central Limit Theorem for simple cases, and employ these results to obtain asymptotic approximations.

5. Derive and assess estimators
   Obtain Method‑of‑Moments and Maximum‑Likelihood estimators, evaluate bias and variance, and determine asymptotic distributions.

6. Construct confidence intervals and conduct hypothesis tests
   Compute/interpret confidence intervals, construct statistical tests

Course contents

1 - Probability and Distribution Theory: Probability foundations: sets, counting & axioms, Conditional probability, Bayes’ rule & independence, Random variables, distribution functions and expectation, Catalogue of standard discrete / continuous distributions, Joint distributions and densities, Independence and dependence of random variables, Conditional distributions and moments, Random vectors

2 - Statistical Inference: Random samples, Sample statistics, Sampling distributions, Modes of convergence of random variables, Law of Large Numbers, Central Limit Theorems

3 - Point and Interval Estimation Theory: Estimators and their finite‑sample properties, Method of Moments, Likelihood and Maximum‑Likelihood Estimation, Interval estimators and confidence regions

4 - Hypothesis Testing: Statistical tests and their operating characteristics, Neyman–Pearson framework, Likelihood‑ratio, Wald and Score tests

Readings/Bibliography

Casella, G. and Berger, R.L. (2002) Statistical Inference. 2nd Edition, Duxbury Press, Pacific Grove.

Abadir, K. M., Heijmans, R. D. H., & Magnus, J. R. (2018). Statistics. Cambridge: Cambridge University Press.

 

Teaching methods

Traditional lectures

Assessment methods

Assessment will be based on take-home problem sets covering the main topics of the course. Students will be required to solve the assigned exercises and submit a PDF document prepared in LaTeX, containing clear and well-organised solutions with the main steps and reasoning.

After submission, each student will take part in a short individual discussion with the instructor. The aim of the discussion is to review the submitted work and give students the opportunity to explain the methods used and clarify any aspects of their solutions.

The assessment will be conducted on a pass/fail basis. A pass will be awarded when the submitted work shows an adequate understanding of the course topics and the student is able to discuss the main ideas and methods used in the solutions.


Use of Artificial Intelligence

Artificial intelligence may be used to support individual study through additional explanations, summaries, self-assessment activities, language revision, and assistance with LaTeX formatting.

For the take-home assignments, limited, declared, and non-substantial use of AI is permitted. AI tools may be used to clarify concepts, discuss possible approaches, or improve the presentation of the work. However, they must not be used to generate complete solutions, proofs, calculations, or explanations to be submitted as the student’s own work.

Any use of AI in preparing the assignment must be briefly acknowledged in the submitted PDF, indicating the tool used and the purpose for which it was employed. Students remain responsible for the accuracy of the submitted material and must be able to explain and discuss every part of their solutions during the individual oral discussion.

Teaching tools

Students will have access to a complete set of course notes prepared by the instructor, the slides used during lectures, and a collection of exercises designed to support the application of the theoretical and analytical tools covered in the course. All materials will be made available through the University of Bologna’s Virtuale platform.

Office hours

See the website of Enzo D'Innocenzo