Ing Ind - Inf (Mag.)(ord. 270) - MI (475) ELECTRICAL ENGINEERING - INGEGNERIA ELETTRICA

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052573 - APPLIED STATISTICS

Obiettivi dell'insegnamento

Thecourseaims to introduce students to statistical data analysis and it is intended to provide basic knowledge of inferential statistics.

In the first part, thecourse will be devoted to the review of the basic probabilistic tools useful for statistical analysis. The core of the course, focused on statistical analysis, will lead the student to be able to understand and interpret basic statistical analyses, with particular attention to engineering problems related to random phenomena, as well to be aware of the limits of the information obtained from the data.

Finally, the students will be trained to deal with real data using the statistical (free) software R. In this part of the course, particular attention will be paid to apply statistical tools to real data appearing in electrical engineering (e.g. measurements from electrical systems). This will help students to understand which kind of problems one can find in dealing with true data and to learn how these difficulties can be practically solved.

Risultati di apprendimento attesi

To describe the "Expected learning outcomes" of the course we refer to "Descrittori di Dublino" (DdD, in the following).

DdD 1: knowledge and understanding.

Lectures and exercise sessions will allow students to:

-understand the basic principles of mathematical statistics and probabilistic modellling;

-learn the appropriate terminology;

DdD 2: applying knowledge and understanding

Lectures and exercise sessions will allow students to:

-apply methods of statistical analysis to specific problems in engineering;

-extract significant indicators from data;

-use statistical softwares;

DdD4: communication skills.

The laboratory project will allow students to:

-motivate the conclusions of a statistical analysis in a convincing way;

-communicate clearly and convincingly the design choices made.

Argomenti trattati

ELEMENTS OF PROBABILITY THEORY. Random vectors: joint distribution functions, joint and marginal probability density functions. Covariance and variance of the sum of random variables.

ESTIMATION AND HYPOTHESIS TESTING. Point and interval estimation: general notions and examples. Sampling distributions from a normal population. Tests of hypotheses: type I and type II errors, critical region, test statistics, size and level of significance of the test, power function, p-value. Tests on the mean and the variance of a normal population. Test on a proportion. Inference for the differences between two normal means, between two proportions and for the ratio between two variances of normal populations.

MULTIPLE REGRESSION and ANOVA. Least square estimation, confidence interval and hypothesis testing concerning the regression coefficients, prediction of a future response, coefficient of determination and adjusted R^2, analysis of residuals: assessing the model. Introduction to the analysis of variance.

The statistical software "R" will be introduced as a tool for data analysis.

Prerequisiti

The student is supposed to have already used univariate probability distributions and to have received preliminary notions of descriptive statistics.

The following arguments are given for known and can be found in the textbook by Ross. The argments will be only reviewed in the first two lectures of the course.

1) Random variables; cumulative distribution function and its properties; discrete and continuous random variables. Jointly distributed random variables and independent random variables. Expected value (or: expectation, mean, mean value) and its properties; expected value of sums of random variables. Variance; variance of sums of independent random variables. [Ross] Chapt. 4, excluded Sects. 4.3.2, 4.7, 4.8, 4.9.

2) Special random variables: Bernoulli and binomial distributions, [Ross] Sect. 5.1 pp. 141-147; Poisson distribution, [Ross] Sect. 5.2 pp. 148-155; uniform distribution [Ross] Sect. 5.4 pp. 160-164; Gaussian or normal distribution, [Ross] Sect. 5.5; Exponential distribution, [Ross] Sect. 5.6 pp. 177-180. Skip the parts on the moment generating functions.

3) Sample mean and sample variance. Sample mean, [Ross] Sects. 6.1, 6.2. Central limit theorem and approximate distribution of the sample mean, [Ross] Sect. 6.3. Sample variance [Ross] Sect. 6.4. Sampling from a normal population; distribution of the sample mean, [Ross] Sects. 5.8.1 (5.8.1.1 no), 5.8.2, 6.5.1.

Modalità di valutazione

Student preparation will be evaluated by a written examination and a data analysis project (laboratory project).

The examination consists in two parts:

-the first part is a written examination with 4 exercises (numerical problems);

[DdD: 1,2]

-the second part is the Lab Project (composed by a short report written by the students and an oral presentation of the project).

[DdD: 1,2,4]

Both parts are mandatory.

The final grade will be a weighted mean of the two grades, 60% for the written examination and 40% for the Lab. Project.

The two parts are descibed in more details here below.

Written examination.

The written examiniation consists in the solution of 4 exercises (numerical problems) concerning the argument described in the program.

In particular exercises will possibly concern:

1) properties of random variables; 2) point estimation (MLE, moments estimators, plug-in estimators)

3) confidence interval and tests (normal sample: mean with known and unknown variance, variance; mean in bernoulli samples: asymptotic intervals; mean for general random variables: asymptotic intervals; two sample problems: difference of means -known variances, unknown equal variances, unknown and unequal variances; Paired t-Test for paired data; variances of two normal populations). 4) goodness of fit and independence in contingency tables.

Laboratory Project

A) groups must be of 2 or 3 persons.

B) The lab. project must be discussed in one of the dates specified at the beginning of the course.

C) Each group need to write a short report (max 3 pages + figures & tables) containing: the description of the dataset, aim of the project, exploratory analysis, inferential analysis, conclusions. Each group must deliver the project on the day of the oral presentation.

D) Each group will present the project with some slides, 10 min for each project plus questions. Important: some question will be asked to each member of the group.

Bibliografia

Sheldon Ross, Introduction to probability and statistics for engineers and scientists, Editore: Elsevier Academic Press, Anno edizione: 2009, ISBN: 9780123704832 Note: