# ELEMENTS OF PROBABILITY

## Learning outcomes of the course unit

[knowledge and understanding]

Know, understand and be able to explain all the essential arguments in the section "Programma esteso" below, which form an essential background of probability and statistics for the applications

[applying knowledge and understanding]

Be able to solve exercises and problems on the course arguments, in particular all the "homeworks" assigned during the lessons and all the exercises of the book [Ross] from chapters 3-8

[making judgements]

Be able to check whether a phenomenon is non deterministic and when it is possible to model it with one of the standard models of random variables presented

[learning skills]

Be able to read and understand scientific texts which build on the knowledge of inferential statistics in one variable

## Prerequisites

Binomial coefficient. Derivation and integration in one variable.

## Course contents summary

Combinatorics. Elementary probability theory. Discrete and continuous models of random variables. Inferential statistics in one variable. Confidence intervals. Classical statistical tests.

## Course contents

Sample space, events and their composition, axioms of probability and properties. Finite spaces with equiprobable elementary events, combinatorics. Conditional probability, total probability formula, Bayes formula. Independent events. Binary tests and classical paradoxes.

Discrete and continuous random variables, probability mass and density functions, cumulative distribution function. Linear and non-linear transformations.

Random vectors, joint law and marginal laws, independence. Sum, min and max of random variables.

Expected value, expectation of a function, linearity. Variance, standard deviation, mode, median, quartiles, range, covariance, correlation. Special formula for the expectation of a non-negative r.v., Markov and Chebyshev inequalities, weak law of large numbers.

Models of discrete random variables: Bernoulli, binomial, Poisson. (Possibly also uniform, geometric, negative binomial and hypergeometric.)

Models of continuous random variables: uniform, exponential, Gaussian. (Possibly also gamma, beta and Weibull.)

Law of the sum of several independent random variables, reproducibility, momoent generating function, weak law of large numbers, central limit theorem.

Randomness in the industrial processes, process control. Accuracy, precision and capability of a process.

Inferential statistics, population, sample, sample statistics, unbiased and consistent estimators. Sample mean, sample variance, distribution in the Gaussian case, chi-squared law.

Confidence intervals, bilateral and unilateral. Auxiliary functions: Gaussian case, estimation of the mean with variance known or unknown, estimation of the variance with mean known or unknown; exponential case, estimation of the parameter; Bernoulli case, estimation of p; two Gaussian samples, estimation of the difference of the means (homoskedastic hypothesis), estimation of the ratio of the variances.

Classical hypothesis testing, bilateral and unilateral. Test performed on the statistics, on the estimator and on the p-value. Operating characteristic curve, industrial language.

## Recommended readings

Francesco Morandin - Lecture notes 2018-2020

Francesco Morandin - Lecture notes 2021 (developed during the course and available online after each lesson)

Sheldon Ross - Introduction to probability and statistics for engineers and scientists

## Teaching methods

Traditional classes and exercise sessions. Arguments are presented in a practical way and formalized only when useful. Much stress is given to the motivations and many examples are presented. Applied exercises and theoretical homework (the latter are optional) are assigned regularly during lessons. During exercise sessions will be presented the solution of some of the exercises and problems assigned in the previous lessons.

## Assessment methods and criteria

The examination is a written test with two probability problems and two statistics problems. The student is required to solve three out of four, in fact all four problems will be graded, but only the best three scores will be added to grade the whole test. (Each problem is worth 11-13 points and it is split in three parts, the first one is quite elementary and worth about 7 points, the other two are more advanced and worth 2-3 points each.) The final score of the test is given by the sum of the three problems with the highest scores, possibly increased by a percentage bonus for early hand in of the exam paper. There is an upper bound to 30 points, with "laude" given if the total was at least 33.

It is possible to ask for an oral examination after the written one, but if the final evaluation is not positive, the student must redo the written part.

To pass the exam the student should master the mathematical language and formalism. He must know the mathematical objects and the theoretical results of the course and he should be able to use them with ease.

## Other informations

There will be an e-learning website, where the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC