Università degli Studi di Urbino Carlo Bo / Portale Web di Ateneo


Probability and Statistics
Probabilità e Statistica Matematica

A.Y. Credits
2014/2015 6
Lecturer Email Office hours for students
Renzo Lupini Mon-Tue from 10:30 am 11:00 am and from 01:00 pm to 01:30 pm.

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.


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.

Teaching, Attendance, Course Books and Assessment


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


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.


Oral exam.

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


The course is offered both face-to-face and on-line within the Laurea Degree Program in Applied Computer Science.

« back Last update: 24/07/2014


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