04642 - Probability Calculus and Statistics

Learning outcomes

At the end of the course the student has achieved a basic knowledge of probability and mathematical statistics. He is able to solve simple problems of probability and statistical inference.

Course contents

Events and random numbers. Operations and relations. Constituents. Logical dependence and independence. Expectation and probability. Conditional expectation and probability. Formulae of composite expectations and probabilities, Bayes formula, formulae of total expectations and probabilities. Correlations between events. Covariance, variance and correlation coefficient. Chebychev inequality. Weak law of large numbers. Stochastically independent events. Stochastically independent events. Stochastically independent random numbers taking a finite number values. Bernoulli scheme and generalized Bernoulli scheme. Binomial, multinomial, geometric and Poisson distribution. Hypergeometric distribution. Convergence of binomial to Poisson distribution. Cumulative distribution function. Probability density. Computation of the expectation of functions of random numbers with absolutely continuous distribution. Uniform, exponential and normal or Gaussian distributions.  Computation of expectation and variance. Estimate of the tails of stantard Gaussian distribution. Convergence of geometric distribution to exponential distribution and of binomial distribution to standard Gaussian distribution (De Moivre-Laplace theorem without proof). Joint cumulative distribution functions and marginal cumulative distribution functions. Joint and marginal probability densities. Stochastic independence. Uniform multi-dimensional distribution function in a region. Multi-dimensional Gaussian distribution. Statistical inference on the parameter of Bernoulli distribution and on the mean and the variance of Gaussian distribution. Markov chains with discrete and continuous time. Poisson process. M/M/1 and M/M/infinity queueing processes.

Readings/Bibliography

F. Biagini, M. Campanino. Elements of Probability and Statistics.An Introduction to Probability with de Finetti's Approach and to Bateian Statistics. Springer.

Additional text for further reading: 
B. de Finetti. Theory of Probability. Wiley.  

Teaching methods

The course is based on lectures in which the basic concepts of Probability Theory, of Mathematical Statistics in the Bayesian approach and of Markov chains with applicatitions to queueing systems will be illustrated. The exposistion of theoretical partnwill be accompanied with examples of applications to computer science and the development of simple exercises in order to familiarize students with the concrete applications of theoretical concepts.

Assessment methods

Final verification is composed by a written and oral test.

Written test consists in the solution of four exercises on probability events and discrete distributions, absolutely continuous distributions, statistical inference, Markov chains with the  goal of verifying the ability of the student of applyng the notions exposed in the course.

Oral exam consists in a talk, starting from three questions, with the goal of testing the understanding of the basic concepts of the course and the ability of developing simple logical arguments.

 

 

 

 

 

 

 

 

Teaching tools

Lectures.

Office hours

See the website of Massimo Campanino