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Department of Mathematics and Applications
"Renato Caccioppoli"


Stochastic Models and Statistical Methods

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Stochastic Models and Statistical Methods

Credits

6

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/06 Probability and Mathematical Statistics.

Examination method

Oral exam.

Learning
objectives

The course aims to introduce students to the study of continuous-time stochastic processes with a discrete state space. Particular attention is paid to birth-death processes and to queueing theory, through the formulation and analysis of mathematical-probabilistic and simulation models suitable for describing real systems. A further objective is to make students aware of the relevant issues involved in constructing stochastic models of physical, biological and economic phenomena and in their statistical analysis, as well as the issues involved in constructing numerical simulations.

Syllabus

Service systems. Little’s laws. Poisson process. Birth-death processes. Random variables of particular interest, including gamma, hyperexponential and chi-squared variables. Markov chains. Ergodicity. Queues: M/M/1, M/M/1/K, M/M/s, M/M/∞, M/D/1, M/G/1, GI/M/s. Queues with Erlang distribution. Introduction to estimation theory and statistical hypothesis testing. Applications of statistical tests. Specific instances of the Monte Carlo method. Simulation of random variables. Simulation of service systems and related statistical analysis. Use of R to implement simulation and analysis algorithms.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

  • know and understand the theoretical foundations of the stochastic models covered and the general issues relating to stochastic modelling and to the development and analysis of stochastic simulation algorithms;
  • are able to apply the knowledge acquired to develop simulation algorithms independently;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

The criteria for assessing learning and for grading are, in order: clarity, correctness and completeness of exposition; ability to develop simulation algorithms.