The Basics of Probability Theory and Statistics
A.Y. 2023/2024
Learning objectives
The course unit focuses on the basic elements of probability, estimation theory and hypothesis testing. Particularly, the Least Squares method and the related hypothesis testing will be discussed in some details.
Expected learning outcomes
The students attending this course unit are expected to acquire the basic concepts of probability and statistics. This is obtained by giving them the key points of the axiomatic theory of probability and some relevant numerical applications of the theory.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Probability according to Laplace and Von Mises. The axiomatic definition of probability. Sample space and event space. Basic theorems on probability. Conditional probability and stochastic independence of events.
Random variables in one dimension. The cumulative distribution function. Mean and variance. The Chebyshev inequality. The two-dimensional random variables: joint and marginal distributions, mean, variance and covariance, linear correlation coefficients. The n-dimensional random variable (basics).
Distribution of functions of random variables. The expectation theorem. Covariance propagation.
Sequences of random variables. The Bernoulli theorem. The central limit theorem
Populations and samples. The histogram. Statistical Inference and Maximum Likelihood. Unbiased and consistent estimators. Sample mean and sample variance. Least Squares. Tests of hypotheses. Tests of hypotheses on Least Squares Estimators.
Random variables in one dimension. The cumulative distribution function. Mean and variance. The Chebyshev inequality. The two-dimensional random variables: joint and marginal distributions, mean, variance and covariance, linear correlation coefficients. The n-dimensional random variable (basics).
Distribution of functions of random variables. The expectation theorem. Covariance propagation.
Sequences of random variables. The Bernoulli theorem. The central limit theorem
Populations and samples. The histogram. Statistical Inference and Maximum Likelihood. Unbiased and consistent estimators. Sample mean and sample variance. Least Squares. Tests of hypotheses. Tests of hypotheses on Least Squares Estimators.
Prerequisites for admission
Calculus and Linear Algebra.
Teaching methods
Mixture of direct instruction and flipped classroom.
Teaching Resources
Mood A.M., Graybill F.A. and Boes D.C., Introduction to the theory of statistics. Mc Graw-Hill, 1974, ISBN: 0-07-042864-6
Papoulis A., Pillai S. U., Probability, random variables and stochastic processes. Mc Graw-Hill, 2002, ISBN: 0-07-366011-6
Cramer H., Mathematical methods of statistics. Princeton University Press, 1957
Papoulis A., Pillai S. U., Probability, random variables and stochastic processes. Mc Graw-Hill, 2002, ISBN: 0-07-366011-6
Cramer H., Mathematical methods of statistics. Princeton University Press, 1957
Assessment methods and Criteria
The final exam consists in an oral discussion organized in questions and answers concerning the topics treated during lectures.
The final assessment will be based on the following criteria: knowledge of the topic treated during the frontal lectures; critical reasoning; skill in the use of specialistic lexicon; analysis of specific probabilistic models.
The final score will be expressed in thirtieth.
The final assessment will be based on the following criteria: knowledge of the topic treated during the frontal lectures; critical reasoning; skill in the use of specialistic lexicon; analysis of specific probabilistic models.
The final score will be expressed in thirtieth.
ICAR/06 - SURVEYING AND MAPPING
MAT/06 - PROBABILITY AND STATISTICS
SECS-S/01 - STATISTICS
MAT/06 - PROBABILITY AND STATISTICS
SECS-S/01 - STATISTICS
Practicals with elements of theory: 24 hours
Lessons: 32 hours
Lessons: 32 hours
Professor:
Barzaghi Riccardo
Professor(s)