This course aims at providing the students with the mathematical tools (set-theoretic and algebraic
structures, differential and integral calculus in one or several real variables, ordinary differential
equations) whose knowledge is indispensable for the achievement of the degree. A particular
attention is paid to the concrete application of the learned notions.
This course aims to provide the students with the fundamental of descriptive statistics, inferential statistics and probability theory.
1) Some notions of set theory.
2) The complete ordered field of the real numbers.
3) Euclidean distance and induced topology on the real line. Absolute value of a real number.
4) Cartesian plane.
5) Real functions of one real variable.
6) Polynomials and polynomial functions. Power, exponential and logarithmic functions. Trigonometric functions.
7) Limit of a function of one real variable.
8) Continuity of a function of one real variable at one point. Fundamental theorems on continuos functions.
9) Derivative of a function. Derivation rules. Fundamental theorems on differentiable functions.
10) Monotonicity of a function. Local and global minima and maxima of a function.
11) Convex functions.
12) Riemann integral. Integration rules. Improper integrals.
13) Ordinary differential equations. The separable and the linear case.
14) The vector space R^n. Geometrical representations of the vectors in R^2 and R^3.
15) Euclidean distance in R^n and induced topology on R^n. The cases n=2 and n=3.
16) Distance between two points in the plane and geometrical loci. Conics.
17) Linear algebra. Matrices and operations on them. Determinant of a square matrix.
18) Functions of more variables. Level curves and level sets.
19) Linear and affine functions. Quadratic forms.
20) Continuity of a function of more variables.
21) Differentiable functions in more variables. Partial derivatives.
22) Local and global minima and maxima of a function of more variables.
Part I) descriptive statistics.
Univariate statistics: main chart (pie chart, bar chart, histogram e box-plot), measures of location (mean, mode and median), measure of spread (range, interquartile range, variance, standard deviation), measure of asymmetry (third moment, skewness index, Pearson's skewness coefficient) measure of kurtosis (fourth moment, kurtosis, excess kurtosis).
Bivariate statistics: main representations (contingency tables e shattered plots), main measures (mean, variance and covariance), correlation analysis (linear regression and Pearson's correlation coefficient).
Part II) Probability theory
Probability: probability definition (classic and modern), event taxonomy (independent events, mutually exclusive events, complementary event, union event and intersection event). Conditional probability. Probability of notable events.
Random variables: discrete random variable (discrete probability distribution, expected value and variance), continuous random variable (probability density function, expected value and variance), main continuous distributions (uniform, gaussian, standard normal and chi-square).main discrete distributions (binomial and Bernoulli), central limit theorem, Chebyshev’s inequality, convergence in law of random variables and limit random variable.
Part III) Inferential Statistics.
Estimation theory: estimation problem, main properties of an estimator (unbiased, consistency and efficiency). point estimation (expected value and variance), interval estimation (expected value and variance).
Hypothesis test: Problem statement (first type and second type error, theoretical distribution), testing process, chi-square based independence test.