2024/2025

Degree | Type | Year |
---|---|---|

2503852 Applied Statistics | FB | 1 |

- Name:
- Carmelo Puliatti
- Email:
- carmelo.puliatti@uab.cat

- Bogdan Vasile Crintea
- Maria Doris del Carmen Potosí Rosero

You can view this information at the end of this document.

The student should have completed the course "Càlcul 1". It is required to have practice in differentiating and integrating one-variable functions.

The objective of this subject is that the student assimilates and learns the concepts and tools in analysis that will be necessary to understand important results in Statistics (least-square minimization, joint probability densities, central limit theorem, simulation of variables, determination of laws through moments or the characteristic function, stochastic equations, etc.). These knowledge is classified into four sections:

1. Complex numbers.

2. Integral transforms.

3. Differential calculus in several variables.

4. Integral Calculus in several variables.

1. Complex numbers.

2. Integral transforms.

3. Differential calculus in several variables.

4. Integral Calculus in several variables.

- KM01 (Knowledge) Recognise the language and basic tools of calculus in one and various variables.
- SM03 (Skill) Solve, using numerical methods, optimisation problems, linear algebra and analysis in general that appear in science and, especially, in statistics.
- SM04 (Skill) Resolve problems associated with the extreme points of functions of one and several variables, and the calculation of moments.

1. Complex numbers.

The imaginary unit. Complex arithmetic .Fundamental theorem of algebra.

Polar form of a compex number, roots. Exponential and logarithmic function.

Differentiation and integration of complex-valued functions.

The imaginary unit. Complex arithmetic .Fundamental theorem of algebra.

Polar form of a compex number, roots. Exponential and logarithmic function.

Differentiation and integration of complex-valued functions.

2. Power series

Taylor's formula. Concept of power series.

Power series expansions. Examples.

Euler's formula, the complex exponential.

3. Improper integrals.

Different kinds of improper integrals. Probability laws. Expected value.

The Gaussian law.

Convergence criteria for positive functions. Convergence criteria for (complex) valued functions.

Laplace transform and characteristic function of a density. The domain of definition, examples.

Moment generating function.

4. Integral calculus in several variables.

Coordinate systems in Euclidean space. Polar, cylindric and spherical coordinates.

Functions of several variables, representation methods (graphics, level sets).

Curves and surfaces, parametric and continuous form.

Riemann sums in several variables. Multiple integrals.

Calculus of integrals: Fubini's theorem and change of variable.

5. Differential calculus in several variables.

Linear approximation at apoint: differential and tangent plane.

Partial derivatives, gradient, the chain rule.

Non constrained optimization.

Implicit functions.

Constrained optimization.

Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|

Type: Directed | |||

Solving problems sessions | 15 | 0.6 | |

Theory class | 30 | 1.2 | |

Type: Supervised | |||

Seminars | 5 | 0.2 | |

Type: Autonomous | |||

Personal Study | 30 | 1.2 | |

Solving problems | 62 | 2.48 |

In the learning process it is fundamental the own work of the student, who at all times will have the help of the professor.

The hours of class are distributed in:*Theory:* The teacher introduces the basic concepts corresponding to the subject, showing examples of their application. The student will have to complement the explanations of the professors with the personal study.*Problems:* By completing sets of exercises, the comprehension and application of the concepts and tools introduced in the theory class is attained . The student will have lists of problems, a part of which will be solved in the problem classes. Students should work on the remaining ones as part of their autonomous work.*Seminars:* to reach a deeper understanding of the subject the students work o in group on more complex practical problems. Some seminars will deal with computer-aid approach to solving problems.

**Annotation**: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|

First mid-term exam | 40% | 2.5 | 0.1 | KM01, SM03, SM04 |

First submission of exercises | 10% | 1.5 | 0.06 | KM01, SM03, SM04 |

Second mid-term exam | 40% | 2.5 | 0.1 | KM01, SM03, SM04 |

Second submission of exercises | 10% | 1.5 | 0.06 | KM01, SM03, SM04 |

A continuous assessment is performed based on:

a) Two written tests combining theory and problems, with grades P1 and P2

b) Submission of two sets of exercises, with grades LL1,LL2. Can be completed at home and uploaded to Campus Virtual.

Submissions in b) are manadatory with no resit assesment.

If both P1,P2 have been attended, a grade C1 is generated according to C1=(0,1)(LL1+LL2)+(0,4)(P1+P2).If C1 is at least 5, the final grade is C1.

In order to pass the subject, the each one of the qualifications P1 and P2 has to be be greater or equal to 3.

Students with C1<5 and having submitted b), and students willing to improve their grade, may attend a resit exam, with grade R.

Single evaluation: see catalan version.

The final grade C2 after the resit exam is C2=(0,1)(LL1+LL2)+(0,8) R.

Student’s assessment may experience some modifications depending on the restrictions to face-to-face activities enforced by health authorities.

The professor in charge will publish studying material in the online campus CV. Besides, at the website

the following books are recommended:

1. M. Brokate, P.Manchanda,A.H.Siddiqi, Calculus for Scientists and Engineers, http://link.springer.com/openurl?genre=book&isbn=978-981-13-8464-6

2. A.I. Khuri, Advanced Calculus with Applications in Statistics, https://onlinelibrary.wiley.com/doi/book/10.1002/0471394882

3. P. Dyke, Two and three dimensional Calculus with applications in science and engineering, https://onlinelibrary.wiley.com/doi/book/10.1002/9781119483731

Other useful references are:

4. A.Reventos, Temes diversos de fonaments de les Matemàtiques, pdf accessible al CV.

S. L. Salas, E. Hille. Cálculo de una y varias variables. Ed. Reverté, 1994.

No software is needed

Name | Group | Language | Semester | Turn |
---|---|---|---|---|

(PAUL) Classroom practices | 1 | Catalan | second semester | afternoon |

(PAUL) Classroom practices | 2 | Catalan | second semester | afternoon |

(TE) Theory | 1 | Catalan | second semester | afternoon |