Fundamental of Mathematics and Statistics

Real numbers. Linear algebra: vectors, matrices, systems of linear equations.
Differential and integral calculus with one variable. Functions and their graphs. Limits. Derivates. Integrals. Rules of derivation and integration. Maxima and minima of functions. Convex functions. Taylor formula. Integrals.

Descriptive statistics. Types of Data. Frequency tables. Pictures of Data. Position indexes: mean, median, mode, percentiles, quartiles. Variation indexes: variance, standard deviation. Probability. Axioms of a probability measure. Classical probability. Combinatorial calculus: counting the possible configurations of n elements in k places. Conditional probability. Total probability theorem. Bayes formula. Finite and continuous random variables (r.v.). R.v. law. Probability function of a finite r.v..
Density and distribution functions of a continuous r.v..
Binomial distributions. Continuous distributions: normal distribution. Mean, variance, standard deviation of a finite r.v. and of a continuous r.v. and their properties. Central limit theorem. Sums of independent random variables and normal approximation. Confidence intervals for the mean of a normal distribution, with standard deviation known and with standard deviation unknown. Binomial proportion confidence interval.Statistical inference: populations and samples, the basic idea of statistical test. Power, protection and the detection of differences: Type I and Type II error. Comparisons for enumeration data: Fisher's exact test, the χ2 test, contingency tables. Comparisons of two sample means: Student's t test. Comparisons of any number of sample means: the Analysis of Variance (ANOVA). One-way ANOVA: the completely random design, the randomised complete block design. Regression analysis: the linear regression model and equation, tests of hypotheses. ANOVA of regression.