Download Probability and Statistics II* – 6 CFU (for Mathematics) E. Di Nardo

Probability and Statistics II* – 6 CFU
(for Mathematics) E. Di Nardo
*This course intends to give complementary topics to the contents of the undergraduated course “Probability
and Statistics I”. In particular, differently from the contents of “Probability and Statistics I”, whose final goal is
a correct analysis of data with the help of statistical inference, in this course, all concepts are formalized
within the theory of measure probability and special enphasis is given to theorems and their proofs.
Sample space. Sigma-algebra. Borel sigma-algebra. Probability measure and its main properties.
Boole inequality. Inclusion and exclusion principle. Event sequences. Lim inf and Lim sup. Limits
of event sequences. Equivalence theorem. Continuity property. Lemma of Borel-Cantelli.
Conditioned probability and its main properties. Events mutually independent. 0-1 law. Bayes
theorem and its generalization. Probability space. Cumulative distribution function and its
properties. Probability distribution function. Probability density. Absolutely continuous random
variables. Mixtures. Riemann-Stieltjes integrals. Decomposition of a cumulative distribution
function.
Bernoulli process and the geometric random variable. Bernoulli sample space. Random walks.
Memoryless property. Pascal random variable. Negative binomial random variable. Poisson random
variable as a limit of a binomial distribution. Poisson process. Poisson process as the limit of the
Bernoulli process. Moments of a random variables. Absolute moments. Mean of functions of
random variables. Variance. Moment generating function. Transformations of random variables:
discrete and continuous distributions. Gamma laws: the exponential random variable, the
memoryless property and its applications in realiability theory. Gamma distribution and its
properties. Connection with the chi-square distribution.
Random vectors. Joint cumulative function. Marginal cumulative function. Random variables equal
almost all. Random variables identically distributed. Indipendent random variables. Conditioned
random variables. Conditioned cumulative function. Conditioned density function. Transformations
of pairs of random variables. Convolutions. Stochastic processes. Trajectories. Markov property.
Chapman-Kolmogorov equation. White noise. Brownian motion. Mixed moments of random
vectors. Covariance. Correlation coefficient. Applications to linear regression. Gaussian vectors.
Some fundamental properties for n=2. Characterizations and the role played by the correlation
coefficient. Fundamental properties and sampling distributions of the multivariate normal
distributions. Central limit theorem and its generalizations. Law of large numbers and its
generalizations.
Conditioned mean and its properties. Conditioned mean from a finite generated sigma-algebra.
Conditioned mean from a more general sigma-algebra. Martingales.
Reference
P.Billingsley Probability and measure III Edizione (1995)