# Mathematics and statistics [Cognomi F-O] (2010/2011)

Course code
4S02690
Credits
12
Coordinator
Teaching is organised as follows:
MATEMATICA 8 MAT/05-MATHEMATICAL ANALYSIS I semestre Corrado Marastoni
STATISTICA 4 MAT/06-PROBABILITY AND STATISTICS I semestre Federico Di Palma

#### Learning outcomes

Module: STATISTICA
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This course aims to provide the students with the fundamental of descriptive statistics, inferential statistics and probability theory.

Module: MATEMATICA
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This course aims at providing the students with the mathematical tools (set-theoretic and algebraic structures, differential and integral calculus in one or several real variables, ordinary differential equations) whose knowledge is indispensable for the achievement of the degree. A particular attention is paid to the concrete application of the learned notions.

#### Syllabus

Module: STATISTICA
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Part I) descriptive statistics.
Univariate statistics: main chart (pie chart, bar chart, histogram e box-plot), measures of location (mean, mode and median), measure of spread (range, interquartile range, variance, standard deviation), measure of asymmetry (third moment, skewness index, Pearson's skewness coefficient) measure of kurtosis (fourth moment, kurtosis, excess kurtosis).
Bivariate statistics: main representations (contingency tables e shattered plots), main measures (mean, variance and covariance), correlation analysis (linear regression and Pearson's correlation coefficient).

Part II) Probability theory
Probability: probability definition (classic and modern), event taxonomy (independent events, mutually exclusive events, complementary event, union event and intersection event). Conditional probability. Probability of notable events.
Random variables: discrete random variable (discrete probability distribution, expected value and variance), continuous random variable (probability density function, expected value and variance), main continuous distributions (uniform, gaussian, standard normal and chi-square).main discrete distributions (binomial and Bernoulli), central limit theorem, Chebyshev’s inequality, convergence in law of random variables and limit random variable.

Part III) Inferential Statistics.
Estimation theory: estimation problem, main properties of an estimator (unbiased, consistency and efficiency). point estimation (expected value and variance), interval estimation (expected value and variance).
Hypothesis test: Problem statement (first type and second type error, theoretical distribution), testing process, chi-square based independence test.

Module: MATEMATICA
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Algebra. Sets, relations and functions. Real numbers. Linear algebra (affine geometry).
Functions of one real variable. Generalities. The topology of the real line, of the extended real line and of affine spaces. Limits, continuity and local behaviour. Derivation. Drawing the graph of a function. Integration.
Functions of several real variables and differential equations. Functions of two or more real variables: generalities, partial derivatives, differential. Ordinary differential equations: the linear case, the separable variables case.

#### Assessment methods and criteria

Module: STATISTICA
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The exam requires two tests. The former test is composed by close questions while the latter is composed by open questions and exercises. To pass the exam, the student have to get a positive mark in both the tests.

Module: MATEMATICA
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Written test

 Author Title Publisher Year ISBN Note D. OLIVIERI Istituzioni di statistica CEDAM 2005 Testo ad un livello di approfondimento superiore a quello presente nel corso Navidi Probabilità e statistica per l'ingegneria e le scienze McGraw-Hill 2011 978-8-88-3866334-5 Testo adatto a chi intende approfondire l'uso della statistica in ambito applicativo. Bland Statistica Medica Apogeo Testo adatto a chi intende approfondire l'uso della staitistica nelle discipline mediche.