28578 - Probability and Stochastic Processes

Learning outcomes

The course aims:

• to give a systematic revision of basic probability theory presented in the first level course;
• to introduce advanced knowledge in probability theory;
• to develop the probabilistic reasoning and its applications.
• to give the basis of the stochastic processes theory and
• to discuss some important stochastic processes and their applications.
• to give the background for  numerical calculations of stochastic processes in discrete and continuous time.

 

 

Course contents

 

• Fundamentals: random events, axioms and probability rules.
• Hints on the philosphy of probability.
• Random variables.
• Limit theorems: convergence criteria, laws of large numbers, central limit theorem.
• Generalities on stochastics processes: basic concepts and examples.
• Stationary process. Ergodic process.
• Markov chains. Theoretical basis and applications.
• Classification of states.
• Recurrance and transience.
• Limit distribution and stationary distribution.
• Continuous-time Markov Chains.
• Transition rate matrix.  Embedded Markov chain.
• Poisson process.
• Waiting times and interarrival times
• Birth-and-Death processes.
• Some ideas about brownian motion.

 

Simulation methods in studying stochastic processes.

• Elements of R. Computer experiments.
• Solving problems through simulation.
• Random walks and variations on the theme.

 

Readings/Bibliography

G.R. Grimmet, D.R. Stirzaker, Probability and Random Processes, Claredon Press, Oxford

W. Feller, 1957, An introduction to probability theory and its applications, Wiley, New York.

S.M. Ross, Stochastic Processes, Wiley, New York.

Notes from the teacher are available (in Italian).

Teaching methods

Classes, computer workshops.

Assessment methods

Written and oral exam

 

Teaching tools

Computer workshop.

Additional readings will be indicated during the lessons.

 

Office hours

See the website of Rodolfo Rosa