The common thread is the concept of function and its role in mathematical modeling to solve problems, introducing the fundamentals of differential and integral calculus and their application to inferential calculus with the study of the statistical significance of the sample regression line.
Numeric sets with operations: exponential, logarithmic and trigonometric numbers. Real power of a positive real number. The absolute value of a real number. Application: Intervals of real numbers and subsets of points of the Cartesian line.
Sets and subsets. Operations between sets: the complement, the union, the intersection, de Morgan's laws, the relative complement, the Cartesian product. Application: The set of results of a discrete and continuous random phenomenon. The set of events of a discrete random phenomenon: the set of parts. The set of events of a continuous random phenomenon: σ-algebras.
Functions. Domain, codomain or image set. Functions and Equations. Injective, surjective, bijective functions. Invertibility of a function. Continuous functions. Application: Functions to measure the probability of a random event. The probability spaces. Discrete and continuous random variables. The discrete case: the probability distribution function. The continuous case: the probability density function.
Derivation and Anti-Derivation. Geometric interpretation of the derivative function and of the primitive function.
The Fundamental Theorem of Integral Calculus (statement). The fundamental formula of integral calculus: the definite integral. Application: The link between the probability distribution function and the probability density function for continuous random variables. The measure of P(a≤X≤b) where X is a continuous random variable.
The main continuous random variables and their properties: Uniform, Gaussian, t-Student's, χ ^ 2, Fisher's F.
A linear model for indirectly measurable quantities: the regression line.
Inferential study of the statistical significance of the sample regression line: estimate, confidence intervals and hypothesis testing for intercept and slope of the regression line.