## PROBABILITY AND STATISTICS PROBABILITA' E STATISTICA MATEMATICA

A.Y. Credits
2015/2016 6
Lecturer Email Office hours for students
Renzo Lupini Thursday and Friday from 11.00 am to 12.00 am.

### Assigned to the Degree Course

Date Time Classroom / Location

### Learning Objectives

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, namely estimation and hypothesis testing.

### Program

01. Discrete univariate random variables:
01.01 Generalities on random phenomena. Quantitative observations of systems in Physics, Economics and Biology. Frequencies of occurrence and law of large numbers.
01.02 Probability distribution and cumulative distribution function.
01.03 Bernoulli and Poisson distributions.
01.04 Events and their probabilities. Conditional probability. Independence and incompatibility.

02. Continuous univariate random variables:
02.01 Probability density and cumulative distribution functions.
02.02 Uniform, Gauss, Gamma and Chi-square distributions.
02.03 Functions of random variables and their probability density functions: modulus, power and square.

03. Analysis of univariate random variables:
03.01 Measures of location and spread: median, average and variance.
03.02 Tchebyshev Theorem.
03.03 Average and variance of Bernoulli, Gauss, Gamma and Chi-square distributions.
03.04 Higher order moments: skewness and kurtosis.
03.05 Moment-generating functions and characteristic functions.

04. Bivariate r.vs:
04.01 Joint and marginal distribution functions.
04.02 Bivariate Bernoulli and Gaussian distributions.
04.03 Events and their probabilities. Conditional probabilities and stochastic independence.
04.04 Real functions of bivariate random variables. Sum of independent random variables.
04.05 Center and Variance-Covariance matrix. Linear regression lines.
04.06 Linear and quadratic functions of bivariate random variables and their distributions.
04.07 Characteristic functions.

05 Multivariate random variables:
05.01 Probability distribution functions of multivariate random variables. Center and Variance-Covariance matrix nxn. Least-squares hyperplane.
05.02 Characteristic functions. Probability distribution functions of sums of independent multivariate random variables.
05.03 Multivariate Bernoulli and Gaussian distributions.
05.04 Univariate random variables associated with systems of normal standard independent random variables: Chisquare, T-student, F-Fisher

06 Sequence of random variables
06.01 Convergence in probability.
06.02 Convergence in probability to a number of a sequence of random variables. convergence in probability of the algebraic means to the average. Law of large numbers and Central Limit Theorem.

07. Statistics
07.01 Populations and samples.
07.02 Sample functions for independent samples and their distributions.
07.03 Estimators of average and variance. Efficiency and Bias.
07.04 Interval estimation.
07.05 Hypotheses testing: test of normality, test of homogeneity and independence.
07.06 Test Chi-square.

### Bridging Courses

Although there are no mandatory prerequisites for this exam, students are strongly recommended to take it after Calculus.

It is also worth noticing that the topics covered by this course will be used in Digital Signal and Image Processing.

### Learning Achievements (Dublin Descriptors)

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.

### Teaching Material

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

### Teaching, Attendance, Course Books and Assessment

Teaching

Theory lectures and exercises, both face-to face and on-line.

Attendance

Although recommended, course attendance is not mandatory.

Course books

Lupini, "Lezioni di Probabilità e Statistica", Quattroventi, 2007.

Baldi, "Calcolo delle Probabilità e Statistica", McGraw-Hill, 1998.

W. Navidi, "Statistics", Mc Graw-Hill.

Assessment

Oral exam.

The oral exam is passed if the mark is at least 18/30.

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.

### Notes

The course is offered also on-line inside the Moodle platform > elearning.uniurb.it

 « back Last update: 17/11/2015

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