04642 - Probability Calculus and Statistics

Learning outcomes

At the end of the course, the student knows basic concepts and methods of probability and mathematical statistics. The student can solve simple problems of probability and statistical inference.

Course contents

○ Mathematical model of a random experiment: sample space, events, axioms of probability and their consequences.

○ Conditional probability and independence: chain rule, total probability rule and Bayes' rule.

○ Combinatorics and discrete uniform probability spaces.

○ Random variables:

• Distribution (or law) and cumulative distribution function.
• Discrete and (absolutely) continuous random variables: discrete and continuous probability density functions.
• Expected value and variance.
• Relevant probability distributions: Bernoulli, binomial, Poisson, discrete and continuous uniform, exponential, normal (or Gaussian).

○ Random vectors:

• Joint law, marginal laws, joint cumulative distribution function, independence of random variables, covariance.
• Discrete random vectors: joint discrete and marginal discrete probability density functions.

○ Descriptive statistics: population and sample, types of data, frequencies, tabular and graphical representations; measures of central tendency, measures of variability.

○ Bivariate data: joint frequencies and two-way tables; scatter plot; covariance and linear correlation coefficient; method of least squares and linear regression.

○ Limit theorems:

• Sequence of i.i.d. random variables.
• Law of large numbers: Chebyshev's inequality, Monte Carlo method.
• Central limit theorem.

○ Discrete-time Markov chains: transition matrix, directed graph representation, n-step transition probability, communication classes, invariant distribution.

Readings/Bibliography

Lectures notes and sheets of exercises prepared by the teacher (available on virtuale.unibo.it in Italian).

Supplementary textbook: Q. Berger, F. Caravella, P. Dai Pra, Probabilità, seconda edizione, Springer, 2021

Teaching methods

Lectures and exercises will be alternated in order to explain theoretical concepts through examples.

Attendance is not compulsory but is strongly recommended, as active participation helps to gain a deeper comprehension of the topics treated in the course.

Lectures are in Italian.

Assessment methods

Learning assessment is done only through a final exam, which is a written test lasting about two hours. The test includes four exercises based on the topics covered during the course. It’s graded out of 32 points (8 points per exercise). If you score 31 or 32, the final grade is 30 with honors (30 e lode). The exercises are similar to the ones in the “exercise sheets” available on virtuale.unibo.it.

Students are not allowed to use books or notes during the test. They can use a calculator and one handwritten A4 sheet with only formulas and theory (no exercises).

It can be asked to take an optional oral exam to improve the grade, but only with a score at least 18/30 on the written test. The final grade will then be the average of the written and oral exam scores.

Teaching tools

Website of the course available on virtuale.unibo.it, where the student can find: lecture notes, sheets of exercises, past written exams, and other useful material for the course.

Office hours

See the website of Elena Bandini