Statistics, Probability and Stochastic Processes
Instance: 2012/2013 - 1S
Study cycles/ courses
||No. of students
||Syllabus since 2011/12
Teaching - Hours
This course aims to endow students with underlying knowledge of Statistics and Probability, which is indispensable to take decisions in uncertainty situations that happen in various areas of Engineering.
This course also aims to endow students with accurate communication skills when themes in the domain of Statistics and Probability are referred. Students will also develop a critical attitude in the analysis of engineering problems and they will be able to apply their knowledge in the resolution of practical problems. The adequate learning of the fundamental concepts of this course will make students able to easily learn advanced knowledge in their future career, both academic and professional.
Probability: Sample spaces, outcomes, and events; Axioms and properties of probability; Conditional probability; Combinatorics and probability.
Discrete random variables: Probabilities involving random variables; Multiple random variables; Expectation; Probability generating functions; The binomial random variable; The weak law of large numbers; Conditional probability and expectation.
Continuous random variables: Densities and probabilities; Expectation; Moment generating functions and characteristic functions; Probability bounds.
Cumulative distribution functions: Continuous, discrete and mixed random variables; Functions of random variables and their cdfs; The central limit theorem.
Statistics: Parameter estimators and their properties; Histograms; Confidence intervals for the mean; Hypothesis tests for the mean; Regression and curve fitting; Monte Carlo estimation.
Bivariate random variables: Joint and marginal probabilities; Conditional probability and expectation; The bivariate normal; Extension to three or more random variables.
Random vectors: Random vectors and random matrices; Linear estimation of random vectors (Wiener filters); Estimation of covariance matrices; Nonlinear estimation of random vectors; Gaussian random vectors;
Random processes: Definition and examples; Strict-sense and wide-sense stationary processes; WSS processes through LTI systems; Power spectral densities for WSS processes; Characterization of correlation functions; The matched filter; The Wiener filter.
Advanced concepts in random processes: The Poisson process; Renewal processes; The Wiener process;
Introduction to Markov chains: Discrete-time Markov chains; Recurrent and transient states; Limiting n-step transition probabilities; Continuous-time Markov chains.
Mean convergence and applications: Convergence in mean of order p; The Karhunen–Loeve expansion; Projections, orthogonality principle, projection theorem; The spectral representation; Convergence in probability; Convergence in distribution; Almost-sure convergence.
John A. Gubner; Probability and random processes for electrical and computer engineers, 2006. ISBN: 9780521864701
Papoulis, Athanasios; Probability, random variables, and stochastic processes . ISBN: 0-07-100870-5
Teaching methods and learning activities
Theoretical-classes: presentation of the themes of the course illustrated by examples, which explain the concepts and results presented;
Theoretical-practical classes: exercises proposed and solved by the professor. Students will be encouraged to actively participate in class by suggesting solutions to the exercises and by criticizing results.
Type of assessment
Distributed evaluation with final exam
Eligibility for exams
Continuous assessment will be based on homeworks with a total weight of 30% of the final grade.
Calculation formula of final grade
Final exam: 70%