# Statistics

Tipo di corso: Lecture
Valore in crediti ECTS: 5

Objective
The course starts with a self-contained review of the basic concepts of probability theory. Discrete and continuous univariate and multivariate random variables are presented and applications to reliability of parallel and series systems are discussed. The properties of expected value, variance and covariance are studied and the use of transform methods to derive the moments of a random variables is explained. Stochastic dependence, conditional distributions and conditional expectation are discussed and computer science applications and some important distributions in the field are presented. The definition of stochastic process with examples in availability analysis is finally introduced and analyzed. The second part of the course focuses on two classical topics of statistics: statistical inference and regression analysis. We cover the basics of parameter estimation theory by studying two important and widely used estimation methods (maximum likelihood and the method of moments) and then move the attention to hypothesis testing. Least-square estimation concludes the course.

Contents
Probability Theory
 Probability Models, sample space, events, probability axioms, combinatorial problems, conditional probability, independence of events.
 Series and parallel systems, reliability of a component/system, product law of reliabilities,parallel redundancy, product law of unreliabilities, law of diminishing returns.
 Discrete random variables, probability mass function, distribution function, special distributions, discrete random vectors, independent random variables.
 Series-parallel systems, series-parallel reliability block diagrams, blocks in series / parallel / kout-of-n.
 Continuous random variables, exponential distribution, reliability and failure rate, important distributions.
 Functions of random variables, jointly distributed random variables, functions of Normal random variables.
 Expectation, moments, transform methods, mean time to failure.
 Conditional distribution and expectation, mixture distributions.
 Law of large numbers, standardisation of a random variable, standardisation of random sums,central limit theorem.
 Stochastic processes, classification of stochastic processes, the Bernoulli process, the Poisson
 process, renewal processes, availability analysis.
Statistical inference
 Parameter Estimation, point estimation problem, estimator, estimate, unbiased estimator
 Maximum likelihood estimation method, likelihood function, log-likelihood, ML estimator.
 Test of hypothesis. Null and alternative hypothesis. Simple and composite hypothesis. Hypothesis testing problem, acceptance region, critical region.
Error of type I and error of type II, level of significance of the test, power function of the test. Neyman-Pearson Lemma.
 Method of least squares, LS estimator, normal equations, variance of the estimator, BLUE estimator (Gauss-Markov Theorem), paramet