Programme

Part I: Probability Theory

1.⁠ ⁠Introduction to probability: sample spaces, events, operations on events, and axioms of probability.
 2.⁠ ⁠Equally likely outcomes and combinatorial counting techniques: arrangements, binomial and multinomial coefficients.
 3.⁠ ⁠Conditional probability, law of total probability, and Bayes' theorem.
 4.⁠ ⁠Independence of events and applications to classical probability problems.
 5.⁠ ⁠Discrete and continuous random variables: probability distributions, cumulative distribution functions, and densities.
 6.⁠ ⁠Random vectors: joint and marginal distributions, independence of random variables.
 7.⁠ ⁠Expected value and its properties: linearity, the probabilistic method, and applications.
 8.⁠ ⁠Variance, covariance, and variance of sums of random variables.
 9.⁠ ⁠Moment-generating functions and their applications.
10.⁠ ⁠Markov's and Chebyshev's inequalities and the weak law of large numbers.
11.⁠ ⁠Fundamental discrete distributions: Bernoulli, binomial, geometric, and Poisson.
12.⁠ ⁠Fundamental continuous distributions: uniform, exponential, normal, chi-square, and Student's t. The Poisson process.

Part II: Descriptive statistics and inference

13.⁠ ⁠Descriptive statistics: frequencies, graphical representations, measures of location and dispersion, percentiles, box plots, and sample correlation.
14.⁠ ⁠Sampling and sampling distributions: sample mean and variance, sampling from finite populations, and normal samples.
15.⁠ ⁠Central limit theorem and asymptotic approximations of sampling distributions.
16.⁠ ⁠Point estimation: method of moments, maximum likelihood, Bayesian estimation, and criteria for evaluating estimators.
17.⁠ ⁠Confidence intervals and hypothesis testing: inference for means and proportions, comparisons between populations, goodness-of-fit and independence tests.
18.⁠ ⁠Simple linear regression: parameter estimation, statistical inference, confidence and prediction intervals, coefficient of determination, and residual analysis.

Prerequisiti

Basics notions from calculus in one real variable

Core Documentation

Sheldon Ross, “Introduction to Probability and Statistics for Engineers and Scientists", Elsevier Science Publishing Co Inc.

Reference Bibliography

Sheldon Ross, “Introduction to Probability and Statistics for Engineers and Scientists", Elsevier Science Publishing Co Inc.

Attendance

Optional.

Type of evaluation

The assessment consists of an initial written examination, 90 minutes long, which must be passed to proceed to the oral examination. The written examination aims to assess students' knowledge of fundamental concepts and their ability to apply them to problem-solving exercises. The subsequent oral examination will focus on evaluating their understanding of theoretical topics, mastery of key concepts, and ability to discuss and elaborate on them independently. In accordance with the Dublin Descriptors, student assessment takes into account the level of knowledge and understanding of the course topics, as well as the ability to apply acquired knowledge, analyze problems, and develop independent reasoning. A passing grade is awarded to students who demonstrate an understanding of the fundamental concepts and the ability to apply them to simple problems. Intermediate grades require a more solid grasp of the course material and a good ability to analyze and solve more complex problems. The highest grades are awarded to students who demonstrate an in-depth understanding of the subject, rigorous reasoning, the ability to synthesize information, and independence in tackling and solving problems, including those arising in contexts not explicitly addressed during the course.

Programme

Part I: Probability Theory

1. Introduction to probability: sample spaces, events, operations on events, and axioms of probability.
2. Equally likely outcomes and combinatorial counting techniques: arrangements, binomial and multinomial coefficients.
3. Conditional probability, law of total probability, and Bayes' theorem.
4. Independence of events and applications to classical probability problems.
5. Discrete and continuous random variables: probability distributions, cumulative distribution functions, and densities.
6. Random vectors: joint and marginal distributions, independence of random variables.
7. Expected value and its properties: linearity, the probabilistic method, and applications.
8. Variance, covariance, and variance of sums of random variables.
9. Moment-generating functions and their applications.
10. Markov's and Chebyshev's inequalities and the weak law of large numbers.
11. Fundamental discrete distributions: Bernoulli, binomial, geometric, and Poisson.
12. Fundamental continuous distributions: uniform, exponential, normal, chi-square, and Student's t. The Poisson process.

Part II: Descriptive statistics and inference

13. Descriptive statistics: frequencies, graphical representations, measures of location and dispersion, percentiles, box plots, and sample correlation.
14. Sampling and sampling distributions: sample mean and variance, sampling from finite populations, and normal samples.
15. Central limit theorem and asymptotic approximations of sampling distributions.
16. Point estimation: method of moments, maximum likelihood, Bayesian estimation, and criteria for evaluating estimators.
17. Confidence intervals and hypothesis testing: inference for means and proportions, comparisons between populations, goodness-of-fit and independence tests.
18. Simple linear regression: parameter estimation, statistical inference, confidence and prediction intervals, coefficient of determination, and residual analysis.

Prerequisiti

Basics notions from calculus in one real variable

Core Documentation

Sheldon Ross, “Introduction to Probability and Statistics for Engineers and Scientists", Elsevier Science Publishing Co Inc.

Reference Bibliography

Sheldon Ross, “Introduction to Probability and Statistics for Engineers and Scientists", Elsevier Science Publishing Co Inc.

Attendance

Optional

Type of evaluation

Assessment will be based on a 90-minutes written examination, typically consisting of 5-6 questions, some theoretical and others more application-oriented, designed to assess both students' understanding of fundamental concepts and their ability to apply them to problem-solving. Each question will be assigned a specific number of points, with the total maximum score generally slightly exceeding 30 points. The subsequent oral examination will focus on evaluating their understanding of theoretical topics, mastery of key concepts, and ability to discuss and elaborate on them independently. In accordance with the Dublin Descriptors, student assessment takes into account the level of knowledge and understanding of the course topics, as well as the ability to apply acquired knowledge, analyze problems, and develop independent reasoning. A passing grade is awarded to students who demonstrate an understanding of the fundamental concepts and the ability to apply them to simple problems. Intermediate grades require a more solid grasp of the course material and a good ability to analyze and solve more complex problems. The highest grades are awarded to students who demonstrate an in-depth understanding of the subject, rigorous reasoning, the ability to synthesize information, and independence in tackling and solving problems, including those arising in contexts not explicitly addressed during the course.