#
Fundamental of mathematics and statistics

A.Y. 2020/2021

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

**Lesson period:**
Activity scheduled over several sessions (see Course syllabus and organization section for more detailed information)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

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.

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

Statistics: the Mathematics part

**Teaching methods**

Streamed lectures and recitations; recorded videos; online discussion forum; self-assessment online tests; programming lab using R

**Teaching Resources**

D. Benedetto et al. Matematica per le scienze della vita. Ambrosiana

M.M. Triola e M.F. Triola, Fondamenti di Statistica (per le discipline biomediche). Pearson

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.

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.

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

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

Lessons: 32 hours

Professors:
Guidone Armando, Rizzo Ottavio Giulio

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

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

Lessons: 16 hours

Professor:
Maggis Marco

Shifts:

Professor:
Maggis Marco

turno 1

Professor:
Maggis Marco### M - Z

Responsible

Lesson period

year

**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.

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

Statistics: the Mathematics part

**Teaching methods**

Streamed lectures and recitations; recorded videos; online discussion forum; self-assessment online tests; programming lab using R

**Teaching Resources**

D. Benedetto et al. Matematica per le scienze della vita. Ambrosiana

M.M. Triola e M.F. Triola, Fondamenti di Statistica (per le discipline biomediche). Pearson

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.

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.

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

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

Lessons: 32 hours

Professors:
Manicone Francescopaolo, Rossini Milvia Francesca

Statistics

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

Lessons: 16 hours

Professors:
Morale Daniela, Ugolini Stefania

Shifts:

Professors:
Morale Daniela, Ugolini Stefania

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