Fundamental of Mathematics and Statistics

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
6.3 Rate of divergence and convergence
6.4 Sequences, series, and recurrence relations
7 — Forecasting future
7.1 Asymptotic behaviour
7.2 Limit computation
7.3 Convergence and divergence speed
7.6 Evolution models with continuous time
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.

Chapter 1 - Number Sets and the Concept of Domain
Chapter 2 - Vectors
2.1 From Numbers to Vectors
2.2 Operations with Vectors
2.3 Vector Direction
2.4 Dot Product
2.5 Linear Systems

Chapter 3 - Matrices and Transformations
3.1 Matrices and Transformations
3.2 Operations with Matrices

Chapter 4 - The Mathematical Form of Natural Phenomena
4.1 Phenomena, Models, Functions
4.2 The Graph of a Function
4.3 Increasing and Decreasing Functions, Maxima and Minima

Chapter 5 - Complex Phenomena and Elementary Functions
5.1 The World of Linear Laws
5.2 Quadratic Laws
5.3 Power Functions and the Dimensions of Life

Chapter 6 - Population Dynamics and Biological Rhythms
6.1 Exponential Functions
6.2 Logarithms

Chapter 7 - Predicting the Distant Future
7.1 The Problem of Asymptotic Behavior
7.2 Calculating Limits
7.3 Rates of Divergence and Convergence

Chapter 8 - The Laws of Change
8.1 Average and Instantaneous Rate of Change
8.2 Rules for Differentiation
8.3 Derivatives: User Guide
8.5 Evolution Models with Continuous Time

Chapter 9 - Integrals
9.1 From Derivative to Function
9.2 Integrating
9.3 Differentiating