Mathematics

- Probability axioms - Discrete probability spaces - Combinatorics - Inclusion-exclusion principle - Independence - Binomial, multinomial and hypergeometric distributions - Conditional probability - Continuity - Random walks, gambler ruin problem. - Discrete random variables: expectation, variance and covariance - Independence of random variables - Bernoulli, binomial, geometric and negative binomial random variables - Sums of independent random variables - Poisson random variable and its binomial approximation - Joint, marginal and conditional densities - Chebyshev's inequality and the weak law of large numbers - Conditional expectation - Continuous random variables: density function, distribution function, expectation and variance - Uniform random variable. Skorohod representation - Exponential random variable - Gaussian random variables and De Moivre-Laplace thorem

The course requires familiarity with topics in the "Algebra Lineare" (Linear Algebra) course and the “Calcolo” (Calculus) course (sets, equivalence classes, functions, integrals, derivatives). This knowledge is indispensable. There are no compulsory propaedeutic courses.

F. Caravenna, P. Dai Pra: Probabilità (Springer) S. Ross: Probabilità. (Apogeo) F. Spizzichino, G. Nappo: Introduzione al calcolo delle probabilità. (Lecture Notes) Suggested exercises will be available.

Theory (60%), Exercises (40%)

Attendance at lessons is recommended, though it is not compulsory.

The examination aims to evaluate learning through a written test (consisting in solving problems similar to the one solved during the lectures) and an oral test (consisting in the discussion of the most relevant topics illustrated during the course). The written test will last about 3 hours and can be substituted by 2 intermediate tests, the first of which will take place mid-course and the second immediately at the end of the course. To pass the exam an evaluation of not less than 18/30 is needed. The student has to prove to have acquired sufficient knowledge in the topics of the program and to be able to perform at least the simplest of the assigned exercises. To achieve a score of 30/30 “cum laude”, the student must instead demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.

A. N. Shiryaev: Probability. W. Feller: An Introduction to Probability Theory and its Applications.

Theory (60%), Exercises (40%)

Random experiments. Outcome space (or sample space). Events and set operations. Read by De Morgan. Definition of probability measure. Uniform measurement on finite outcome spaces and combinatorics. Properties and first theorems relating to the probabilities of unions of events and of inclusion between events. Conditional probability: law of composite probabilities; total probabilities. Bayes' theorem. Independence of two events and in a family of events. Discrete random variables. Discrete probability distributions and discrete probability density (or mass function). Notes on the distribution function. Two-dimensional random variables. Joint densities of two random variables defined on the same space and marginal densities. Conditional distributions. Independence between random variables and inheritance of the independence property for transformations of random variables. Characterization of independence for discrete random variables via discrete probability density. Expected value and linearity properties. Conditional expected value. Theorem on the expected value of the conditional expected value. Variance and covariance. Variance obtained from the conditional expected value. Calculation of variance for the sum of random variables. Correlation coefficient and its properties. The existence of the k-th moment implies the existence of the lower order moment. Measurement continuity theorem (without proof). Special random variables: Bernoulli, binomial, uniform, hypergeometric, geometric, Pascal (or negative Bernoulli) random variable. For all the random variables mentioned, the expected value and variance were calculated. No memory for the geometric random variable. Minimum among independent geometric random variables is still a v.a. geometric. Independent letter sequences and geometric random variables. Poisson random variable: definition and first properties. Approximation of the Binomial via the Poisson. Independent Poisson sum. Sum of an independent Bernoulli number N ∼ Poi(λ) with parameter p. Distribution of a Poisson conditional on the Poisson sum (connection with the binomial). Probability generating function and 1moment generating function or Laplace transform (notes of their use without complete proofs). Markov inequality. Chebyshev inequality. Weak law of large numbers. Uniform random variable (on an interval) and Monte Carlo method to calculate integrals (as seen only for integrals in one dimension). Notes on the Gaussian distribution. Sum of independent Gaussians. Use of Gaussian tables and standardization of a random variable. Outline of the Central Limit Theorem. Use of the normal approximation through two types of exercises. Statistics. Definition of estimator. Unbiased estimators. Unbiased estimator for mean. Unbiased estimator for variance in cases of known or unknown mean. Definition of confidence interval. Linear regression (outline without proof).

The prerequisites are the mathematics that is taught in upper secondary schools and the mathematics that students have learned in the first semester of the first year of university studies in the Degree in Mathematics course.

INTRODUCTION TO PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIENTISTS. Author Sheldon Ross CALCOLO DELLE PROBABILITA' 2/ED Author Paolo Baldi

Attending the course is strongly recommended to have greater involvement both in learning the theoretical part and in solving the proposed exercises.

To pass the exam, students will have to pass a written test which consists of exercises and problems to be solved using the knowledge and skills developed during the course. Those who pass the written test can request to take the oral test to improve their grade and will have to explain the theoretical aspects presented in class.

INTRODUCTION TO PROBABILITY AND STATISTICS FOR ENGINEERS AND SCIENTISTS. Author Sheldon Ross CALCOLO DELLE PROBABILITA' 2/ED Author Paolo Baldi

The lessons are separated into a theoretical part followed by examples and exercises relating to the proven theorems. The written exercises part will be encouraged by giving weekly exercises on which students can practice and which will subsequently be corrected in class.

- Probability axioms (3h) - Discrete probability spaces (3h) - Combinatorics (6h) - Inclusion-exclusion principle (4h) - Independence (6 h) - Binomial, multinomial and hypergeometric distributions (3h) - Conditional probability, Product formula, Bayes formula (5 h) - Continuity of Probability (2 h) - Occupation numbers: Maxwell Boltzmann, Bose-Einstein, Fermi-Dirac (4h) - Discrete random variables: expectation, variance and covariance (6h) - Independence of random variables (2h) - Bernoulli, binomial, geometric and negative binomial random variables (6h) - Sums of independent random variables (2h) - Poisson random variable and its binomial approximation (2h) - Joint, marginal and conditional densities (4h) - Transformation of discrete random variables (2h) - Chebyshev's inequality and the weak law of large numbers (4h) - Conditional expectation and its Geometric interpretation (5h) - Continuous random variables: density function, distribution function, expectation and variance (4h) - Uniform random variable in (0,1). Skorohod representation (3h) - Exponential random variable as limit of Geometric random variables (2h) - Gaussian random variables (2h) -Transformation of continuous random variables (1h) - De Moivre-Laplace thorem and Gaussian Approximation (2h)

The course requires familiarity with topics in the "Algebra Lineare" (Linear Algebra) course and the “Calcolo” (Calculus) course (sets, equivalence classes, functions, integrals, derivatives). This knowledge is indispensable. There are no compulsory propaedeutic courses.

NOTE Changes are possible : the other teacher will be the winner of a competion, F. Spizzichino, G. Nappo: Introduzione al calcolo delle probabilità. (Notes will be available in the e-learning site ) Berger, Caravenna, Dai Pra: Probabilità (Springer) (for students of Sapienza University a free pdf version is available link https://link.springer.com/book/10.1007/978-88-470-4006-9) Other notes and exercises will be available on "Sapienza" e-learning con Moodle.

Theory (60%), Exercises (40%). If necessary, due to sanitary rules, lessons (or part of them) will be online. Notes and exercises will be available on "Sapienza" e-learning con Moodle.

Attending lessons is not compulsary, but it is recommended

The examination aims to evaluate learning through a written test (consisting in solving problems similar to the one solved during the lectures) and an oral test (consisting in the discussion of the written examination and the most relevant topics illustrated during the course). If the student pass the written examination (with at least 18/30) he/she is admitted to the oral exam. If necessary, due to sanitary rules, the written test (or part of it) can be substituted by an homework to be discussed during the oral examination. To pass the exam an evaluation of not less than 18/30 is needed: this evaluation is computed on the basis of the written examination (50%) and the oral test (50%). The student has to prove to have acquired sufficient knowledge in the topics of the program and to be able to perform at least the simplest of the assigned exercises. To achieve a score of 30/30 “cum laude”, the student must instead demonstrate that he has acquired excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent manner.

S. Ross: Probabilità. (Apogeo) A. N. Shiryaev: Probability. W. Feller: An Introduction to Probability Theory and its Applications.

Theory (60%), Exercises (40%). If necessary, due to sanitary rules, lessons (or part of them) will be online. Further details, notes and exercises will be available on "Sapienza" e-learning con Moodle.