28635 - Applied Mathematics T-A

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

Basic notions of combinatorics.
Foundations of probability theory. Events and sets. Customary criteria for probability estimate (classical approach, frequency approach).
Kolmogorov's axioms. Joint probability, conditional probability, independence. Total probability theorem and Bayes' theorem.
Random variables. Discrete and continuous random variables. Cumulative distribution function. Continuous random variables with probability density. Characteristic numerical values of random variables: expected value (mean), variance, standard deviation, mean square error, moments. Pairs and vectors of random variables: joint and marginal cumulative distribution functions,joint and marginal probability densities. Laws of conditional distribution, independence. Characteristic numerical values: mean values, covariance matrix, moments. Correlated and uncorrelated random variables.
Models of random variables. Bernoulli scheme. Binomial, Poisson, uniform, normal, exponential random variables. Relationships among some of these kinds of random variables.
Functions of random variables. Characteristic numerical values: representation of the expected value and of the variance, with applications to some notable cases (sum and product of two random variables, linear combination of a finite number of random variables, case of independent, identically distributed random variables, etc.). Notions on the determination of the probability distribution for a function of one or more random variables.
Limit theorems in probability. Sequences of random variables and notions of convergence. Markov inequality, Chebyshev inequality. Laws of large numbers. Central limit theorem.
Stochastic processes. Definition and classification. Poisson, Wiener process, Markov chain. Continuity, differentiation, integration. Spectral density. Linear filtering.

Readings/Bibliography

H. Hsu, Probabilità, variabili casuali e processi stocastici, ed. McGraw-Hill Italia

Teaching methods

Standard lectures held by the teacher alternating with exercise sessions

Assessment methods

The exam consists in a multiple choice test covering both theory and applications.

Teaching tools

Blackboard and projector.

Office hours

See the website of Carlo Alberto Bosello