Double Degree in International Computer Engineering and Management of Business and Technology

Statistics and Mathematical Analysis

The subject provides an advanced view of the mathematical tools that a technological degree requires. It consists of three chapters: differential equations, multivariable functions, and probability and statistics. In the differential equations chapter the formation, use and resolution of ordinary differential equations is studied; in the chapter of multivariable functions, limits, partials derivatives differentials, equations with partial derivatives, variable changes, gradients, conditioned maximum and minimum and multiple integration (double and triple integrals; and in the probability and statistics chapter, probability spaces are studied such as random variables, functions and operations defined by them and the most typical discreet and continuous distribution functions. Finally, the sampling theory and the statistical hypothesis testing are studied.
Type Subject
Tercer - Obligatoria

Titular Professors

Previous Knowledge

Differential and Integral one variable functions calculus. Vectors spaces and their basic properties.


Chapter 1: Differential equations

1 Introduction
1.1 Definitions
1.2 Classification of differential equations
1.3 Kinds of solutions. Initial value and limit value.
2. Ordinary first order differential equations.
2.1 Separable variables.
2.2 Homogeneous and Reducible to homogeneous.
2.3 Linear.
2.4 Bernoulli.
2.5 Exact differential equations.
3. Superior order differential equations.
3.1 Second order differential equations.
3.2 Second order linear differential equations with constant coefficient.
3.2.1 Homogeneous.
3.2.2 Non homogeneous.
3.3 Second order linear differential equations with variable coefficients.
3.4 Applications.

Chapter 2: Multivariable functions and multiple integrals.

1. Definitions
2. General concepts
2.1 Definition and domain.
2.2 Limits in two variable functions.
2.3 Continuity of a two variables function.
2.4 Graphics, curves and level surfaces.
2.5. Partial and total increase of a function.
2.6 Differential of a function.
2.7 Differentiability.
3. Partial derivative.
3.1 Two variable functions.
3.2 Geometrical interpretation.
3.3 Generalization with more than two variables functions.
3.4 Superior order partial derivatives. Schwartz theorem.
4. Composed functions derivatives.
4.1 Derivative of a composed function. Chain rule.
4.2 Derivation using a tree structure.
5. Implicit functions.
5.1 A variable´s implicit functions. Implicit derivation.
5.2 Various variables implicit functions. Implicit derivation.
6. Gradient and directional derivative.
6.1 Directional derivative: geometrical definition and interpretation.
6.2 Gradient: definition and properties.
6.4. Tangent plane and differential.
7. Maximums and minimums of a multivariable function.
7.1 Necessary conditions for the existence of an extreme.
7.2 Sufficient conditions for the existence of an extreme.
7.3 Tied maximums and minimum. Lagrange multipliers.
8. Double integrals.
8.1 Domain.
8.2 Properties of double integrals.
8.3 Calculus of the double integral. Applications.
8.4 Polar Coordinates
8.5 Variable change. Jacobians.
8.6 Wrapped surface
9. Triple integrals.
9.1 Domain.
9.2 Posing of the triple integral.
9.3 Cylindrical coordinates.
9.4 Variable change. Spherical coordinates.

Chapter 3: Probability and Statistics.

1. Introduction.
1.1 Combinatory
1.2 Concept of probability.
1.3 Axiomatic definition of probability.
1.4 Conditioned probability and independent events.
1.5 Law of total probability and Bayes´s theorem
2. Random variable.
2.1 Definition of a random variable.
2.2 Discreet random variable.
2.3 Continuous random variable.
2.4 Probability density function.
2.5 Distribution function.
2.6 Mathematical expectation and moments.
2.7 Moment-generating function.
2.8 Chebyshev´s theorem.
3. Discrete distributions.
3.1 Bernouilli distribution
3.2 Binomial distribution.
3.3 Geometrical distribution.
3.4 Negative binomial distribution.
3.5 Poisson distribution.
4. Continuous distributions.
4.1 Uniform distribution.
4.2 Exponential distribution
4.2 Normal distribution.
5. Bivariate distributions.
5.1 Discrete distributions
5.2 Continuous distributions
5.3 Joint probability functions.
5.4 Marginal distributions.
5.6 Independent random variables.
5.7 Conditional distributions.
5.8 Covariance and correlation
5.9 Linear regression between two variables
6. Sampling theory. Estimation
6.1 Random sampling
6.2 Estimators. Properties.
6.3 Law of large numbers
6.4 Central limit theorem
6.5 Sample size
6.6 Estimation methods
6.6.1 Moments methods
6.6.2 Estimation by the maximum resemblance method


See the electronic folder of the subject.

Evaluation Criteria

See the electronic folder of the subject.

Basic Bibliography

1. Càlcul diferencial i integral.Daniel Cabedo, Lluís Vicent. 1996.Edicions La Salle. Barcelona.
2. Problemes de Matemàtiques. Daniel Cabedo, Lluís Vicent. 1996 .Edicions La Salle. Barcelona
3. Probabilitat (amb aplicacions a l'estadística). Ramon Villalbí. Lluís Vicent. 2000. Edicions La Salle. Barcelona.
4. Matemáticas superiores para ingeniería. C.R.Wylie. 1994.McGraw-Hill.
5. Cálculo diferencial e integral. N.Piskunov. 1994. Montaner y Simon S.A Barcelona.

Additional Material

1. Lliçons de Càlcul de Probabilitats. Marta Sanz. 1995.Publicacions Universitat de Barcelona.
2. Problemas de Probabilidades y Estadística. C.M. Cuadras. Ediciones PPU. 1990. Barcelona.
3. Problemas de Análisis Matemático. Bombal. Marín. Vera. Editorial AC, libros científicos y técnicos. Madrid.