. – Mathematical statistics I
PROFESSOR LUCIO BERTOLI BARSOTTI
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COURSE AIMS
The general aim of the course is to introduce students to the concept of probability
and lay the basis for its calculation with a view to developing the main statisticalinferential paradigms.
COURSE CONTENT
• Probabilistic model structure
Probability space and events. Random experiment and probability space. Sets of
subsets of an assigned set. Event algebras. Borel sigma algebra. Kolmogorov's
axiomatization and its more direct consequences. Stochastic independence and
conditional probability. Total probability theorem and Bayes' theorem. Subjectivist
approach: betting, odds ratios and odds ratios presented on a logarithmic scale
(logit); problem of coherence; and Dutch book. Frequency and subjectivist
interpretations of probability.
• Univariate random variables
Cumulative distribution function and types of random variable. Calculation of the
probability of events based on the cumulative distribution function. Discrete
variables. Expectation operator (E). Linearity of the expectation operator. Central
and non-central moments. Variance. Standardization. Higher-order moments.
Factorial moments. Equivalence relations between central, non-central and
factorial moments. Discrete random variables of significant practical interest.
Continuous variables: general case. Density (density function) and cumulative
distribution function. Direct moment calculation. Continuous random variables of
significant practical interest. Families of random variables.
• Characteristic function
Computation and properties. Use to calculate moments and identify random
variables. Verification of reproducibility of random variables.
• Transformations and approximations to univariate random variables
Transformations (one-to-one in places) of random variables. Functionals on the set
of distribution functions. Chebyshev inequality. Introduction to stochastic
dominance and functional relationships that preserve second-order stochastic
dominance (Schur-convexity). Jensen inequality. Stochastic convergence:
convergence in probability; convergence in distribution. Sum, product and
transformations of stochastically convergent sequences. The law of large numbers.
The central limit theorem and applications. Asymptotically normal sequences.
Limit distribution of transformations of asymptotically normal sequences.
READING LIST
N. WEISS, Calcolo delle Probabilità, Pearson PBM, 2008.
L. BERTOLI-BARSOTTI, Statistica. Aspetti storici ed assiomatizzazione, ISU-Università Cattolica,
Milan, 1995.
L. BERTOLI-BARSOTTI, Problemi e complementi di calcolo delle Probabilità ed inferenza statistica,
ISU-Università Cattolica, Milan, 1996.
L. BERTOLI-BARSOTTI, Corso di Statistica Matematica, Quaderni del Dipartimento di Matematica,
Statistica, Informatica e Applicazioni Università di Bergamo, Serie Didattica, n. 3, 2005.
A.M. MOOD - F.A. GRAYBILL - D.C. BOES, Introduzione alla Statistica, Mc Graw-Hill, 1991.
TEACHING METHOD
Lectures and exercises.
ASSESSMENT METHOD
Written and oral examination.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board.
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