PROBABILITY AND STATISTICS
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- GIACOMO DIMARCO
- Credits
- 9
- Didactic period
- Secondo Semestre
- SSD
- MAT/06
Training objectives
- This is a course of 72 hours (9 credits; 48 hours of probability and 24 hours of statistics); it takes place in the second didactic period.
The first goal of the course is to bring the student to familiarize with the mathematics of random events, especially through problems, letting him become able to reduce a real problem to a mathematical model and then solve it autonomously, using the theorems studied.
The second goal of the course, not for importance but only for temporal order, is to introduce the first elements of descriptive and inferential statistics. Indeed, the joint use of techniques coming from the two disciplines, probability and statistics, is widely applied in the interpretation and resolution of various problems coming from various contexts: economy, biology, medicine, pharmacology, quality control in productive processes, demography.
The main knowledge acquired will be:
- knowledge of probability measures and their main features
- knowledge of the rules of combinatorics
- knowledge of the concept of conditional probability
- knowledge of the concept of random variable, density, distribution function, mean and variance
- knowledge of the most famous random variables, both discrete and absolutely continuous
- knowledge of the main limit theorems: laws of large numbers and central limit theorem
- knowledge of the main techniques of descriptive statistics for a graphic representation of a set of data
- knowledge of the main techniques of descriptive statistics to compute positional indices, variability indices
- knowledge of the concept of sample distribution of a statistic, estimator, efficiency and non-distortion
- knowledge of the main techniques of inferential statistics: linear regression, statistical tests of hypothesis
- knowledge of the "R" environment for for statistical analysis
The main skills acquired by the students will be to know how to model real problems in the language of probability and statistics, to represent and synthesize sample information and to apply inferential techniques to support decisions.
Specifically, students will:
- be able define a probability measure and apply its properties
- be able to operate with the conditional probability, in particular with the alternatives law and the Bayes formula
- know how to count the elements of a finite set by the techniques of combinatorics
- recognize random variables, how to operate with random variables, compute density and distribution (both joint and marginal), mean and variance
- know how to deal with a Bernoulli process and make predictions about the process
- know how to deal with a Poisson process and make predictions about the process
- know how to apply the limit theorems in various contexts
- to summarize the sample information through graphs and summarize it with indices and interpret the results
- be able to quantify the linear correlation between two numerical variables
- be able to apply the method of linear regression to a set of observed data
- know how to test an hypothesis
- be able to use the environment "R" for statistical analysis Prerequisites
- Two (annual) courses on mathematical Analysis
Course programme
- The program of the course is the following:
Combinatorial analysis. The basic principle of counting. Permutations, combinations. Multinomial coefficients. The number of integer solutions of equations. (4 hours)
Elements of Probability. Sample space and events. Axioms of probability. Sample spaces having equally likely outcomes. Conditional probability and independence. Bayes formula. (8 hours)
Discrete Random Variables. Discrete density. Expected value. Expectation of a function of a random variable. Variance. Bernoulli and Binomial random variables. Poisson random variable. Geometric random variable. Negative binomial random variable. Hypergeometric random variable. Expected value of the sum of random variables. (8 hours)
Continuous Random Variables. Density, distribution function, expectation and variance of continuous random variables. Functions of continuous random variables. Uniform random variable. Normal random variable. Exponential random variable. Approximation of a binomial by a normal random variable. Distribution of a function of a random variable. (10 hours)
Jointly distributed Random Variables. Joint distribution functions. Independent random variables. Sum of independent random variables. Sum of normal random variables. Conditional distributions. Joint probability distribution of functions of random variables. (8 hours)
Properties of Expectation. Simple properties. Expectation of the sum of random variables. Moments of the number of events that occur. Covariance, variance of sum and correlation. Conditional expectation. Moment generating functions. (5 hours)
Limit Theorems. Markov inequality, Chabyshev inequality. Weak and strong law of large numbers. The central limit theorem. (5 hours)
- Descriptive statistics (6 hours)
Graphical representation of data: histograms, frequency polygons, diagrams with circular sectors, bar charts, box plot. Calculation of synthesis positional measures (mean, median, mode, quartiles), variability measures (range, interquartile range, variance, standard deviation, coefficient of variation) and shape (symmetry).
- Sampling distributions (2 hours).
Sampling distribution of a statistic, estimator and estimation, efficiency and no distortion.
- Statistical tests (10 hours).
General principles. Confidence intervals. Z tests and t tests with a sample and with two samples, one-tailed and two-tailed samples, p-value.
- Linear regression (6 hours).
The regression line and the scatter diagram. The linear correlation coefficient. The regression model and parameter estimation. The coefficient of determination. The residue analysis. Tests on the parameters and on the goodness of the model. Didactic methods
- Theoretical and practical lessons. The exercises are carried out togheter with students. In particular, the instructor requires attending students to propose solutions to the exercises.
At the end of every man argument of statistics, there will be some computer exercises using the environment "R" for statistical analysis. During these exercises, the ICT tools necessary to apply the statistical methodologies to some real data sets will be introduced. Learning assessment procedures
- The exam consists of two parts:
- a written examination of 3 hours, where the student is asked to solve some exercises of probability and statistics;
- an oral exam, where the student is asked to describe som arguments of the course, also stating and/or proving some theorems.
The student gets a mark between 0 and 32 for the written examination; he can go to the oral exam if the mark is at least 15/30. The final mark depends on both the examinations and is given by an overall valutation of the student.
The student can apply several times the written exam; when the student gives to the teacher a new manuscript, however, it means that he wants to give up the grade achieved in the previously delivered manuscript. Reference texts
- Probability: lecture notes and exercises come mainly from the following books (in alphabetic order):
- Baldi, Paolo: Calcolo delle probabilità, seconda edizione, McGraw Hill, 2011
- Caravenna, Francesco, Dai Pra, Paolo: Probabilità. Un'introduzione attraverso modelli e applicazioni, Springer, 2013
- Dall'Aglio, Giorgio: Calcolo delle probabilità, terza edizione, Zanichelli, 2003
- Ross, Sheldon M.: Calcolo delle probabilità, seconda edizione, Apogeo, 2007
Statistics: lecture notes and exercises come mainly from the following books (in alphabetic order):
- N. Freed, S. Jones, T. Bergquist, S. Bonnini: Statistica per le scienze economiche e aziendali, 2019, Isedi
- Bonnini Stefano, Grassi Angela: Esercizi svolti di Statistica e Calcolo delle Probabilità, 2015, Voltalacarta
