Fundamental of Mathematics and Statistics
A.Y. 2026/2027
Learning objectives
The course aims to develop students' fundamental knowledge and skills required to describe, interpret, and analyse mathematical models of natural phenomena, as well as to use the main tools of probability and statistics for data analysis and the quantification of uncertainty.
Expected learning outcomes
Upon successful completion of the course, students will be able to describe, interpret, and analyse simple mathematical models of natural processes. They will also be able to use the main tools of probability and statistics to model random phenomena, analyse data, and critically interpret the results of statistical analyses.
Lesson period: year
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
year
Course syllabus
Mathematics module
1. Introduction to number systems
2. Vectors
2.1 From numbers to vectors
2.2 Vector operations
2.3 Direction of vectors
2.4 Dot product
2.5 Systems of linear equations
3. Matrices and transformations
3.1 Matrices and transformations
3.2 Matrix operations
4. The mathematical form of natural phenomena
4.1 Phenomena, models, and functions
4.2 Graphs of functions
4.3 Increasing and decreasing functions, maxima and minima
5. Complex phenomena and elementary functions
5.1 Linear models
5.2 Quadratic models
5.3 Power functions and the dimensions of life
6. Population dynamics and biological rhythms
6.1 Exponential functions
6.2 Logarithms
7. Predicting long-term behaviour
7.1 Asymptotic behaviour
7.2 Computing limits
7.3 Rates of divergence and convergence
7.4 Sequences and their limits
8. The mathematics of change
8.1 Average and instantaneous rates of change
8.2 Rules of differentiation
8.3 Derivatives: applications and interpretation
8.4 Continuous-time models of change
9. Integrals
9.1 From derivatives to functions
9.2 Integration
9.3 Differentiation
Statistics module
Descriptive Statistics
1. Random phenomena. Variables, samples, parameters, and estimates.
2. Empirical distributions: frequencies and their graphical representation. Bar charts and histograms.
3. Data summarisation: descriptive statistics. Measures of central tendency and dispersion. Relative position measures: quartiles and box plots.
4. Correlation and linear regression.
Measuring uncertainty: Probability and random variables
5. The probabilistic model: probability spaces and the basic properties of probability.
6. Independence and conditional probability. The Law of Total Probability and Bayes' Theorem.
7. Modelling with random variables. Parameters characterising probability distributions: expected value, variance, and standard deviation.
8. Discrete random variables: Uniform, Bernoulli, Binomial, and Poisson distributions. Continuous random variables: Uniform and Normal distributions.
9. Fundamental theorems of Statistics: the Law of Large Numbers and the Central Limit Theorem. Normal approximation to the Binomial distribution.
Confidence intervals and hypothesis testing
10. Statistical estimation through confidence intervals: estimation error. One-sample confidence intervals for a proportion, a population mean, and a population variance. Comparison of samples.
11. Statistical hypothesis testing. Null and alternative hypotheses, test statistics, critical regions, significance levels, critical values, one- and two-tailed tests, and p-values.
12. Hypothesis tests for one or two proportions and comparison of one or two population means (independent or paired samples).
13. Non-parametric tests: chi-square tests for independence and goodness of fit.
1. Introduction to number systems
2. Vectors
2.1 From numbers to vectors
2.2 Vector operations
2.3 Direction of vectors
2.4 Dot product
2.5 Systems of linear equations
3. Matrices and transformations
3.1 Matrices and transformations
3.2 Matrix operations
4. The mathematical form of natural phenomena
4.1 Phenomena, models, and functions
4.2 Graphs of functions
4.3 Increasing and decreasing functions, maxima and minima
5. Complex phenomena and elementary functions
5.1 Linear models
5.2 Quadratic models
5.3 Power functions and the dimensions of life
6. Population dynamics and biological rhythms
6.1 Exponential functions
6.2 Logarithms
7. Predicting long-term behaviour
7.1 Asymptotic behaviour
7.2 Computing limits
7.3 Rates of divergence and convergence
7.4 Sequences and their limits
8. The mathematics of change
8.1 Average and instantaneous rates of change
8.2 Rules of differentiation
8.3 Derivatives: applications and interpretation
8.4 Continuous-time models of change
9. Integrals
9.1 From derivatives to functions
9.2 Integration
9.3 Differentiation
Statistics module
Descriptive Statistics
1. Random phenomena. Variables, samples, parameters, and estimates.
2. Empirical distributions: frequencies and their graphical representation. Bar charts and histograms.
3. Data summarisation: descriptive statistics. Measures of central tendency and dispersion. Relative position measures: quartiles and box plots.
4. Correlation and linear regression.
Measuring uncertainty: Probability and random variables
5. The probabilistic model: probability spaces and the basic properties of probability.
6. Independence and conditional probability. The Law of Total Probability and Bayes' Theorem.
7. Modelling with random variables. Parameters characterising probability distributions: expected value, variance, and standard deviation.
8. Discrete random variables: Uniform, Bernoulli, Binomial, and Poisson distributions. Continuous random variables: Uniform and Normal distributions.
9. Fundamental theorems of Statistics: the Law of Large Numbers and the Central Limit Theorem. Normal approximation to the Binomial distribution.
Confidence intervals and hypothesis testing
10. Statistical estimation through confidence intervals: estimation error. One-sample confidence intervals for a proportion, a population mean, and a population variance. Comparison of samples.
11. Statistical hypothesis testing. Null and alternative hypotheses, test statistics, critical regions, significance levels, critical values, one- and two-tailed tests, and p-values.
12. Hypothesis tests for one or two proportions and comparison of one or two population means (independent or paired samples).
13. Non-parametric tests: chi-square tests for independence and goodness of fit.
Prerequisites for admission
There are no specific prerequisites for this course. It is an introductory-level course.
Teaching methods
Attendance at both course modules is optional but strongly recommended.
The course will combine lectures with interactive teaching activities, including group work on problems and questions of various types, as well as tutorial sessions devoted both to more technical exercises and to the problem-solving and modelling aspects of the course.
The main reference for course materials is the textbook. Any additional materials will be made available to students on MyAriel.
The course will combine lectures with interactive teaching activities, including group work on problems and questions of various types, as well as tutorial sessions devoted both to more technical exercises and to the problem-solving and modelling aspects of the course.
The main reference for course materials is the textbook. Any additional materials will be made available to students on MyAriel.
Teaching Resources
For the Mathematics module
Benedetto, D., Degli Esposti, M., & Maffei, C. (2014). Dalle funzioni ai modelli. Il calcolo per le Bioscienze. Casa Editrice Ambrosiana.
The course materials consist of the textbook and the lecture slides. Additional materials will be made available to students throughout the course via MyAriel.
For the Statistics module
Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. E. Probabilità e statistica per ingegneria e scienze. Strumenti e applicazioni in R. 9th ed. Pearson.
Recommended textbook for exercises:
Monti, A. C. (2024). Statistica. Esercizi svolti. Pearson
Benedetto, D., Degli Esposti, M., & Maffei, C. (2014). Dalle funzioni ai modelli. Il calcolo per le Bioscienze. Casa Editrice Ambrosiana.
The course materials consist of the textbook and the lecture slides. Additional materials will be made available to students throughout the course via MyAriel.
For the Statistics module
Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. E. Probabilità e statistica per ingegneria e scienze. Strumenti e applicazioni in R. 9th ed. Pearson.
Recommended textbook for exercises:
Monti, A. C. (2024). Statistica. Esercizi svolti. Pearson
Assessment methods and Criteria
Learning is assessed through separate examinations for the Mathematics and Statistics modules. The results of the two examinations contribute to a single final grade.
The results of the written examinations will be published on the SIFA platform via the UNIMIA portal.
The final grade, expressed on a 30-point scale, is calculated as the weighted average of the grades obtained in the two modules, according to their respective ECTS credits.
Mathematics module
Assessment consists of the following components:
Part A, consisting of questions covering the course syllabus (exercises and problems requiring the description, interpretation, and explanation of mathematical concepts and their applications);
Part B, consisting of a mathematical modelling problem.
The two parts may be taken either during the same examination session or in two separate examination sessions, provided that both are completed within the examination sessions scheduled at the time the course is offered (i.e., by February of the following academic year). After that deadline, students must take both parts again.
In November, students will have the opportunity to take a mid-term test covering part of the syllabus. This test may partially replace Part A. In this case, during the final examination students will only be required to complete the remaining questions of Part A and/or Part B.
The purpose of Part A is to assess students' knowledge and understanding of the topics covered in lectures and tutorials, as well as their ability to apply this knowledge to solve the proposed questions. Assessment criteria include conceptual understanding, correctness of the mathematical procedures, and the accuracy and effectiveness of the language used to communicate the solution.
The purpose of Part B is to assess students' ability to solve a mathematical modelling problem by making appropriate use of the mathematical tools introduced during the course. Assessment criteria include independent judgement, particularly with regard to the effectiveness of the model contextualisation and interpretation, the conceptual appropriateness of the mathematical methods employed, the correctness of the solution, and the quality of its presentation (clarity and conciseness).
An oral examination may be required by the instructor, if deemed necessary for the final assessment, in addition to Parts A and B.
During both Parts A and B, students may use a calculator and a formula sheet prepared by themselves.
Statistics module
Assessment for the Statistics module consists of a written examination and, where deemed necessary, an oral examination.
In the written examination, students are required to answer several theoretical questions and solve three exercises in Probability and/or Statistics. An oral examination may be requested by the instructor if considered necessary for the final assessment or at the student's request.
Assessment is intended to verify students' ability to use the fundamental tools of probability and statistical analysis to extract information from experimental data and quantify uncertainty. Evaluation takes into account the correct use of statistical terminology, the understanding of the methods employed, and the correctness of the results.
During the examination, students may use a calculator and statistical tables.
The results of the written examinations will be published on the SIFA platform via the UNIMIA portal.
The final grade, expressed on a 30-point scale, is calculated as the weighted average of the grades obtained in the two modules, according to their respective ECTS credits.
Mathematics module
Assessment consists of the following components:
Part A, consisting of questions covering the course syllabus (exercises and problems requiring the description, interpretation, and explanation of mathematical concepts and their applications);
Part B, consisting of a mathematical modelling problem.
The two parts may be taken either during the same examination session or in two separate examination sessions, provided that both are completed within the examination sessions scheduled at the time the course is offered (i.e., by February of the following academic year). After that deadline, students must take both parts again.
In November, students will have the opportunity to take a mid-term test covering part of the syllabus. This test may partially replace Part A. In this case, during the final examination students will only be required to complete the remaining questions of Part A and/or Part B.
The purpose of Part A is to assess students' knowledge and understanding of the topics covered in lectures and tutorials, as well as their ability to apply this knowledge to solve the proposed questions. Assessment criteria include conceptual understanding, correctness of the mathematical procedures, and the accuracy and effectiveness of the language used to communicate the solution.
The purpose of Part B is to assess students' ability to solve a mathematical modelling problem by making appropriate use of the mathematical tools introduced during the course. Assessment criteria include independent judgement, particularly with regard to the effectiveness of the model contextualisation and interpretation, the conceptual appropriateness of the mathematical methods employed, the correctness of the solution, and the quality of its presentation (clarity and conciseness).
An oral examination may be required by the instructor, if deemed necessary for the final assessment, in addition to Parts A and B.
During both Parts A and B, students may use a calculator and a formula sheet prepared by themselves.
Statistics module
Assessment for the Statistics module consists of a written examination and, where deemed necessary, an oral examination.
In the written examination, students are required to answer several theoretical questions and solve three exercises in Probability and/or Statistics. An oral examination may be requested by the instructor if considered necessary for the final assessment or at the student's request.
Assessment is intended to verify students' ability to use the fundamental tools of probability and statistical analysis to extract information from experimental data and quantify uncertainty. Evaluation takes into account the correct use of statistical terminology, the understanding of the methods employed, and the correctness of the results.
During the examination, students may use a calculator and statistical tables.
MATH-01/B - Mathematics Education and History of Mathematics - University credits: 8
MATH-03/B - Probability and Mathematical Statistics - University credits: 4
MATH-03/B - Probability and Mathematical Statistics - University credits: 4
Exercises: 48 hours
Practical exercises with elements of theory: 48 hours
Lessons: 40 hours
Practical exercises with elements of theory: 48 hours
Lessons: 40 hours
Professor(s)
Reception:
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