Fundamental of mathematics and statistics

A.Y. 2021/2022
12
Max ECTS
120
Overall hours
SSD
MAT/01 MAT/02 MAT/03 MAT/04 MAT/05 MAT/06 MAT/07 MAT/08 MAT/09
Language
Italian
Learning objectives
The aim of the course is to provide a basic knowledge of the mathematics needed in the natural sciences, and the tools of descriptive and inferential Statistics, together with concepts of probability on which they are based
Expected learning outcomes
At the end of the course students will be able to describe, interpret and explain simple mathematical models describing natural phenomena, also through statistical methods
Course syllabus and organization

A - L

Responsible
Lesson period
year
Synchronous and asynchronous distance learning. Classes will be recorded and published on the Ariel platform
Course syllabus
Mathematics
2 - Vectors
2.1 From numbers to vectors
2.2 Vector operations
2.3 The direction of vectors
2.4 Scalar product
2.5 Linear Systems
3 - Matrices and transformations
3.1 Matrices and transformations
3.2 Matrix operations
4 — Giving a mathematical shape to natural phenomena
4.1 Phenomena, models, functions
4.2 Function graphics
4.3 Increasing and decreasing functions; maxima and minima
5 — Complex phenomena and elementary functions
5.1 Linear functions
5.2 Quadratic functions
5.3 Power functions and the dimensions of life
6 — Population dynamics
6.1 Exponential functions
6.2 Logarithms
7 — Forecasting future
7.1 Asymptotic behaviour
7.2 Limit computation
7.3 Convergence and divergence speed
8 — The laws of change
8.1 Mean and instantaneous variation rate
8.2 The rules of derivatives
8.3 Derivative: book of instruction
8.5 Continuous time evolutionary models
9 — Integrals
9.1 From derivative to functions
9.2 Integration
9.3 Differentiation

Statistics
Descriptive Statistics.
1) Population, sample, parameter, statistics. Types of data and variables. Sampling.
2) Graphs and tables. Frequency tables. Histograms/bar graphs.
3) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.
Probability and random variables.
4) Introduction.Events and space of events; probability of an event.
5) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.
6) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.
7) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.
8) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.
Confidence intervals and Hypothesis tests.
9) Confidence interval for a proportion.
10) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.
11) Confidence interval for the variance of a population normally distributed. Chi-square distribution.
12) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.
13) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.
14) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.
Linear dependence.
15) Linear correlation and hypothesi test on the correlation coefficient.
16) Linear regression.
Prerequisites for admission
Mathematics: Pre-calcalcus high school level mathematics
Statistics: the Mathematics part
Teaching methods
Lectures, flipped classroom (with pre recorded videos), problem solutions, recitations; assessment tests; programming lab using R
Teaching Resources
MATHEMATICS
D. Benedetto et al. Matematica per le scienze della vita. Ambrosiana

STATISTICS
Probability and Statistics for Engineers and Scientists, 9th Edition. Pearson

Or, especially for exercises:
M.M. Triola e M.F. Triola, Fondamenti di Statistica (per le discipline biomediche). Pearson
Assessment methods and Criteria
Written exam for each module, on a range out of 30.
Mathematics: semiclosed exercises and reality problems on the topics developed during the course. The evaluation is aimed at verifying the understanding of the topics and how they can be applied to real problems.
Statistics: exercises and theoretical questions on the topics developed during the course. The evaluation ranges out of thirty and is aimed at verifying the understanding of the theoretical notions and their application to real data analysis.
The duration of the written test is commensurate with the number and structure of the assigned exercises, but in any case will not exceed 3 hours.

Results will be communicated on the SIFA through the UNIMIA portal.
The final mark, out of 30, is given by the average weighted according to the CFU of each module.
Mathematics
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals with elements of theory: 48 hours
Lessons: 32 hours
Statistics
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals with elements of theory: 24 hours
Lessons: 16 hours
Professor: Maurelli Mario

M - Z

Responsible
Lesson period
year
Synchronous distance learning.
Course syllabus
Mathematics
2 - Vectors
2.1 From numbers to vectors
2.2 Vector operations
2.3 The direction of vectors
2.4 Scalar product
2.5 Linear Systems
3 - Matrices and transformations
3.1 Matrices and transformations
3.2 Matrix operations
4 — Giving a mathematical shape to natural phenomena
4.1 Phenomena, models, functions
4.2 Function graphics
4.3 Increasing and decreasing functions; maxima and minima
5 — Complex phenomena and elementary functions
5.1 Linear functions
5.2 Quadratic functions
5.3 Power functions and the dimensions of life
6 — Population dynamics
6.1 Exponential functions
6.2 Logarithms
7 — Forecasting future
7.1 Asymptotic behaviour
7.2 Limit computation
7.3 Convergence and divergence speed
8 — The laws of change
8.1 Mean and instantaneous variation rate
8.2 The rules of derivatives
8.3 Derivative: book of instruction
8.5 Continuous time evolutionary models
9 — Integrals
9.1 From derivative to functions
9.2 Integration
9.3 Differentiation

Statistics
Descriptive Statistics.
1) Population, sample, parameter, statistics. Types of data and variables. Sampling.
2) Graphs and tables. Frequency tables. Histograms/bar graphs.
3) Mean, modal value, median, midrange and their relations. Range, standard deviation, variance and their relations. Percentiles, quartiles and outliers. Boxplot. Weighted mean.
Probability and random variables.
4) Introduction.Events and space of events; probability of an event.
5) Probability of the union and the intersection. Complemento of an event. Independence. Conditional probability. Bayes Theorem.
6) Random Variables. Expected value, variance and deviation standard of discrete r.v.s.
7) Discrete r.v.s: Binomial and Poisson. Continuous r.v.s: Uniform and Normal.
8) Sample distributions. Centrale Limit Theorem. Normal approximation of the binomial distribution.
Confidence intervals and Hypothesis tests.
9) Confidence interval for a proportion.
10) Confidence interval for the mean, and known/unknown variance. T-Student Distribution.
11) Confidence interval for the variance of a population normally distributed. Chi-square distribution.
12) Hypothesis tests:general concepts. Null and alternative hypothesis, test statistic, critical region, level of significance, critical values, one/two tails test, P-value, errors of the first/second kind, power of a test.
13) Hypothesis test for a proportion. Hypothesis test for one sample: test on the mean (known/unknown variance), test on the variance or on the standard deviation.
14) Inference for two independent samples: inference on two proportions. Inference on two means, either for independent samples or for coupled samples.
Linear dependence.
15) Linear correlation and hypothesi test on the correlation coefficient.
16) Linear regression.
Prerequisites for admission
Mathematics: Pre-calcalcus high school level mathematics.
Statistics: Pre-calcalcus high school level mathematics.
Teaching methods
Classroom lectures; online discussion forum; self-assessment online tests; programming lab using R.
Teaching Resources
D. Benedetto et al. Matematica per le scienze della vita. Ambrosiana
R.E. Walpole - R.H. Myers - S.L. Myers - K.E. Ye
Probabilità e statistica per ingegneria e scienze. Strumenti e applicazioni in R · 9/Ed. · con MyLab
Pearson
o anche
M.M. Triola e M.F. Triola, Fondamenti di Statistica (per le discipline biomediche). Pearson
Assessment methods and Criteria
Mathematics: exercises and reality problems on the topics developed during the course. The evaluation is aimed at verifying the understanding of the topics and how they can be applied to real problems.

Statistics:
written exam: exercises and theoretical questions on the topics developed during the course, both during lessons and exercises (also with R).
Oral exam: discussion of some fundamental topics of the course.
The evaluation of the students is aimed at verifying the understanding of theoretical notions and the student's ability to apply them to real data.
The duration of the written test is two and a half hours.

Results will be communicated on the SIFA through the UNIMIA portal.

Each of the two parts, Mathematics and Statistics, are evaluated out of thirty. The final evaluation of the exam is an average weighted with the CFU of the two individual outcomes.
Mathematics
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals with elements of theory: 48 hours
Lessons: 32 hours
Professor: Rigoli Marco
Statistics
MAT/01 - MATHEMATICAL LOGIC - University credits: 0
MAT/02 - ALGEBRA - University credits: 0
MAT/03 - GEOMETRY - University credits: 0
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 0
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 0
MAT/06 - PROBABILITY AND STATISTICS - University credits: 0
MAT/07 - MATHEMATICAL PHYSICS - University credits: 0
MAT/08 - NUMERICAL ANALYSIS - University credits: 0
MAT/09 - OPERATIONS RESEARCH - University credits: 0
Practicals with elements of theory: 24 hours
Lessons: 16 hours
Professor(s)
Reception:
Monday 14-16 by appointment by email; other days by appointment by email
online meeting
Reception:
upon appointment via email
Via Saldini 50, second floor
Reception:
Tuesday, 3–4 pm
Office
Reception:
Please write an email
Room of the teacher or online room