The course is meant to provide the basics of the theory of probability, random variables and distribution functions as well as the main concepts of inferential statistics and hypothesis testing.

01. Probablity calculus:
01.01 Probability space, events.
01.02 Conditional probability. Independence.
01.03 Law of total probability (and its proof).  
01.04 Bayes rule (and its proof).
01.05 Examples, problems and applications. 

02. Discrete random variables:

02.01 Probability Mass function. Independence. Expected value, variance and their properties.

02.02 Special discrete distributions: Bernoulli, binomial and Poisson distributions. 
02.03 Poisson distribution as an approximation for binomial: theorem and proof.
02.04 Geometrical distributions and negative binomial distributions.

03. Continuous random variables:
03.01 Probability density function and distribution function. Independence. Expected value, variance and their properties.
03.02 Special continuous distributions: Uniform distribution, Exponential distribution, Gaussian distributions, Chi-squared, the t-distribution, the F-distribution.

04.  Limit theorems:
04.01 Markov inequality (and its proof). 
04.02 Law of large numbers (and  its proof). 
04.03 Central limit theorem.

05. Statistical Inference:
05.01 Random samples.
05.02 Consistent and unbiased estimators. 
05.03 Sample mean and variance.
05.04 Normal samples. 
05.05 Maximum Likelihood Estimation.

06.  Hypotheses testing:
06.01 Hypotheses testing for mean and variance of a normal sample.  Hypotheses testing for mean and variance of normal  independent samples. 
06.02 Confidence intervals: confidence intervals for mean and variance of a normal sample.
06.03 Goodness of fit test.
06.04 Test of independence.  

Calculus (strongly suggested, not mandatory).

Knowledge and understanding: the student will be acquainted with the basis of the mathematical theory of Probability and of the Inferential Statistics.

Applying knowledge and understanding: the student will be able to theoretically analyse problems where stochastic variability plays a fondamental role.

Making judgements: the student will be able to choose among several approaches the suitable solution to probabilistic problems.

Communication skills: the student will be able to communicate probabilistic informations by use of the techniques of the differential and integral calculus.

Learning skills: the student will learn the methodology to be used in the mathematical formulation of empirical phenomena.

The teaching material prepared by the lecturer in addition to recommended textbooks (such as for instance slides, lecture notes, exercises, bibliography) and communications from the lecturer specific to the course can be found inside the Moodle platform › blended.uniurb.it

Handouts, excercises, and solutions of exams are available in the Moodle platform for blended learning.

Teaching

Theory lectures and exercises.

Innovative teaching methods

The classroom lectures will be integrated with exercises individually and in groups.

Attendance

Course attendance is not mandatory.

Course books

A. Lanconelli, "Introduzione alla Teoria della Probabilità", Independently published.

S. Ross: ”Probabilità e statistica per l’ingegneria e le scienze”, APOGEO. 

S. Ross: ”Probabilità e statistica per l’ingegneria e le scienze”, APOGEO. 

S. Ross: "Probability and statistics for engineers and scientists", Elsevier. 

Assessment

The expected learning outcomes  will be assessed through a written exam which includes exercises and open questions on the topics of the program of the course. The time available to answer the questions proposed is two hours.

The evaluation criteria are: the level of mastery of knowledge, the degree of articulation of the answer, the degree of adequacy of the explanation, the degree of use of mathematical tools, the degree of accuracy of the analysis and the use of any explanatory examples.

Each of the criteria is assessed on the basis of a four-level scale of values/judgments (insufficient, sufficient, good, excellent) with particular weight assigned to the level of mastery of knowledge, the degree of articulation of the response and the adequacy of the explanation.

The mark of the written exam is expressed on a scale from 18 (minimum to pass the exam) to 30 (excellence).

After passing the written exam (18/30), the student can, if he/she wishes, take a supplementary oral exam. The final evaluation will consist of the evaluation obtained on the written paper together with the evaluation obtained in the oral.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.

Teaching

As for attending students.

Attendance

As for attending students.

Course books

As for attending students.

Assessment

As for attending students.

Disability and Specific Learning Disorders (SLD)

Students who have registered their disability certification or SLD certification with the Inclusion and Right to Study Office can request to use conceptual maps (for keywords) during exams.

To this end, it is necessary to send the maps, two weeks before the exam date, to the course instructor, who will verify their compliance with the university guidelines and may request modifications.