34321 - Applied Mathematics M

Learning outcomes

A sound theoretical basis as well as a working knowledge of the fundamental mathematical methods aimed at coping with uncertainty in physical and other phenomena.

Course contents

Probability calculus: introduction; deterministic and random experiments; sample spaces and events; the algebra of events; the various interpretations of probability; axioms of probability; the measure of probability.

Combinatorial analysis: the basic principle of counting; simple permutations; simple dispositions; permutations with repetitions; dispositions with repetitions; cyclic permutations; sampling; binomial coefficients and multinomial coefficients; simple combinations; combinations with repetitions; binomial theorem.

Conditional probability: definition; theorem of total probability; Bayes's formula; independent events.

Random variables: definitions of random variables; distribution function of probability; cumulative distribution function; density function; expected value; variance; Chebyshev's inequality.

Distribution of probability: Bernoulli distribution; binomial distribution; geometric distribution; negative binomial distribution; hypergeometric distribution; Poisson distribution; discrete uniform distribution; continuous uniform distribution; exponential distribution; Rayleigh distribution; gamma distribution; Gaussian distribution; Chi-squared distribution; Student distribution; characteristic functions.

Multiple random variables: definitions; distribution function; joint and marginal probability density functions; conditional distribution functions; independent random variables; means, covariance, moments; correlation.

Functions of random variables: Representation of the expected value and variance with applications to some important cases (sum and product of two random variables; linear combination of an arbitrary number of random variables; independent random variables with the same probability density function). Probability density function for functions of one or more random variables with applications to the sum of two random variables.

Limit theorems: Laws of large numbers and limit theorems; convergence of sequences of random variables; weak laws of large numbers; the central limit theorem; applications of the central limit theorem.

Introduction to Statistics: Introduction; statistical distributions; elementary theory of samples; estimation theory; least squares method.

Introduction to Random Processes: The general concept of a random process, distribution and density functions, autocorrelation function, stationary processes, wide sense stationary processes, independent increment processes, Markov chains, Markov processes, Poisson processes.

Introduction to R: Introduction to the software R for statistical analysis of data: installation  and configuration of R; data manipulation; accessing R packages; writing functions; basics of R programming. 

Readings/Bibliography

- Sheldon M. Ross, “Probabilità e statistica per l'Ingegneria e le Scienze”, Apogeo (II edizione)

- Hwei Hsu, “Probabilità, variabili casuali e processi stocastici”, McGraw Hill Italia

Both textbooks are available also in English:

- Sheldon M. Ross, “Introduction to probability and statistics for engineers and scientists”, Academic Press (4th edition)

- Hwei Hsu, “Probability, random variables, and random processes”, Schaum's Outline Series, McGrow Hill (2nd edition)

Teaching methods

Standard lectures in which the basic theory is explained will be combined with saveral examples and exercises.

Assessment methods

A written test (2 hours) and an oral examination are foreseen.

The written test is mainly focused on exercises for the solution of which the student is expected to apply the theory learned during the course. If the evaluation of the written test is satisfactory, the student has access to the oral examination. The validity of the written test is limited to the same session of exams. The oral examination is aimed at verifying the knowledge gained by the student concerning mainly the theoretical part of class. The final mark takes into account the evaluations of both the written and the oral part of the exam.

Teaching tools

Notebook or tablet PC and projector

Office hours

See the website of Andrea Mentrelli