Unit PROBABILITY AND STATISTICS II

Course
Mathematics
Study-unit Code
A002325
Curriculum
Matematica per l'economia e la finanza
Teacher
Andrea Capotorti
CFU
9
Course Regulation
Coorte 2022
Offered
2022/23
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa integrata

PROBABILITY AND STATISTICS II - PROBABILITY

Code A002326
CFU 3
Teacher Irene Benedetti
Teachers
  • Irene Benedetti
Hours
  • 21 ore - Irene Benedetti
Learning activities Affine/integrativa
Area Attività formative affini o integrative
Academic discipline MAT/06
Type of study-unit Obbligatorio (Required)
Language of instruction Italian.
Contents Convergences of sequences of random variables - Characteristic functions - Classical limit theorems.
Reference texts 1) J. Jacod and P. Protter, Probability Essentials, Springer-verlag Berlin And Heidelberg Gmbh & Co. Kg, 2004.

2) Alan F. Karr, Probability, Springer Science & Business Media, 1993.

Further teaching material, such as updated notes from the lecturer and solving of the proposed exercises, is available on Unistudium.
Educational objectives Deep knowledge of the fundamental concepts of Probability theory, with particular attention to the main results concerning the different types of convergence of sequences of random variables and the characteristic functions, and the classical limit theorems. The student must be able to present, link and compare the main concepts and results presented in the course and to prove the fundamental theorems within the exam program. She/he will have to know how to solve problems, following the example provided by the exercises carried out in classroom.
Prerequisites To better understand the course and pass the exam, the formative goals of the course Probability and Statistics I are mandatory.
Teaching methods Lectures on all the topics of the program; public resolution of exercises to train students to face explicit problems and to explain them.
Other information 1) Attendance: optional.

2) For students with Specific Learning Disorders and/or Disabilities please refer to the web page: http://www.unipg.it/disabilita-e-dsa
Learning verification modality The evaluation includes an oral exam aimed to ascertain the knowledge level and the understanding capability acquired by the student on theoretical and methodological contents as indicated on the program. The oral exam will also test the student presentation skills and her/his autonomy in the organization and exposure of the theoretical topics.

For info on support services for students with Specific Learning Disorders and/or Disabilities please refer to the web page: http://www.unipg.it/disabilita-e-dsa
Extended program Convergence of sequences of random variables: convergence in distribution, convergence in probability, convergence on average r-th, almost sure convergence. Characteristic functions: definition, properties, relation with convergence in distribution, continuity theorems and applications. Classical limit theorems: the laws of large numbers and the central limit theorem.

PROBABILITY AND STATISTICS II - MATHEMATICAL STATISTICS

Code A002327
CFU 6
Teacher Andrea Capotorti
Teachers
  • Andrea Capotorti
Hours
  • 42 ore - Andrea Capotorti
Learning activities Affine/integrativa
Area Attività formative affini o integrative
Academic discipline MAT/06
Type of study-unit Obbligatorio (Required)
Language of instruction Italian
Contents Sufficiency and likelihood principles. Bayesian and non-Bayesian parameter estimators: detection and properties. Theretical aspects of hypothesis testing.
Reference texts G. Casella, R.L. Berger, Statistical Inference, second edition, Thomson Learning, 2002.
Educational objectives Deep knowledge of statistical mathematical principles, both Bayesian and not.Students will be able to face and solve theoretical problems about parameter estimation, hypothesis testing and analysis of variance. They will be also able to consciously express the learned notions.
Prerequisites To be able to understand the course and to pass the examination are mandatory the formative goals of the course Probability and Statistic I
Teaching methods Lectures on all the subjects and public resolution of exercises to train students to face explicit problems and to explain them.
Other information For students with Specific Learning Disorders and/or Disabilities please refer to the web page: http://www.unipg.it/disabilita-e-dsa
Learning verification modality Written (with possibility of partial intermediate testing during lectures) of two exercises apt to verify ability in facing explicit problems and oral examination of around half an hour apt to verify the study on all the subjects and the ability in explaining and manipulate the studied notions.

The result of the written test is not binding to access the oral test but constitutes the starting point and has a weight of about 1/4 on the final result.
Extended program Transformation of random variables: monotonic and generic transformations; integral probability transformation; order statistics; conditional expexted value and variance.Sufficient and Likelihood Principles: sufficient principle; sufficient, minimal sufficient, ancillary and complete statistics; likelihood principle; MLE estimators and their invariance property.Finding and evaluating classical and Bayesian point estimators: moments method; unbiasedness; efficiency; Cramer-Rao lower bound; UMVUE estimators.Theoretical Deepening on Hypothesis Testing: LRT tests, power function.
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