Course Details for A.Y. 2013/2014

Course Objectives

Students should develop a certain familiarity with the basic probabilistic tools and should become able to - model simple real problems and to propose a solution - solve theoretical problems in secrete and continuous basic probability, using the appropriate mathematical tools. - understand the main probabilistic structures to be able to employ them even in more complex situations; - read a basic probability textbook; - gain access to a more advanced probability course

Course Content

• Probability  spaces: combinatorics, axiomatic approach, uniform probability spaces, conditional probability and independence
• Discrete random variables Distributions: Bernoulli, binomial, Poisson, Hypergeometric, geometric, Joint distributions and independence Transforms of random variables, distributions of max, min, sums of random variables. Expectation, expectation of a function of a r.v., moments, variance and covariance, correlation, lineare regression Examples and applications
• Continuous r.v.'s Distribution functions: definitions and properties. Main distributions : uniform, exponential, Gaussian, Gamma, Beta. Distributions of transform, of  max and min . Expectation, moments, variance and covariance. Joint distributions, sums of independent r.v.'s, conditional distributions. Conditional expectation and conditional variance Moment generating functions or characteristic functions Multivariate Gaussian laws.  
• Convergence of r.v.'s and approximation. The law of large numbers, the central limit theorem, normal approximation.
• If enough time An introduction to Markov chains: recurrence and transience, invariant probabilities,  ergodic theorem.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

Fundamentals of Calculus are strongly recommended

Assessment Methods and Criteria

Written and possibly oral exams.

Textbooks

• P. Baldi, Calcolo delle probabilità , McGraw-Hill.    
• D. Ross, Calcolo dlle probabilità , Apogeo. testo consigliato    
• Schaum's otuline series : Probabilità testo consigliato    

Course page updates

• A.Y. 2012/2013
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• A.Y. 2019/2020