Multidimensional Distributions
Code: 104857 ECTS Credits: 6| Degree | Type | Year | Semester |
|---|---|---|---|
| 2503852 Applied Statistics | OB | 2 | 1 |
Contact
- Name:
- Antoni Sintes Blanc
- Email:
- antoni.sintes@uab.cat
Teaching groups languages
You can check it through this link. To consult the language you will need to enter the CODE of the subject. Please note that this information is provisional until 30 November 2023.
Prerequisites
Elementary probability. Real random variables. Differential and integral calculation. Elemental algebra: vector spaces and determinants.
Objectives and Contextualisation
The probability distribution of a random vector (discrete or continuous) is the main objective of this course.
We analyze the principal characteristics of the joint distribution: the vector of means, the variance-covariance matrix, the
marginal and conditional distributions, etc.
As a principal example we study the multidimensional normal distribution. It is an important continuous distribution, with applications in linear
model theory, multivariate analysis and statistical decision theory.
Theory and calculations related to continuous multidimensional distributions, strongly depends on integral and differential calculus
with functions of several variables. Having this in mind, we review the principal technics focusing on multivariate probability
applications.
Competences
- Correctly use a wide range of statistical software and programming languages, choosing the best one for each analysis, and adapting it to new necessities.
- Make efficient use of the literature and digital resources to obtain information.
- Select statistical models or techniques for application in studies and real-world problems, and know the tools for validating them.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
Learning Outcomes
- Make effective use of references and electronic resources to obtain information.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Use software to visualise data with multi-dimensional distributions.
- Use statistical software to obtain summary indices of the variables in the study.
- Use the properties of the functions of distribution and density.
Content
1. Random vectors.
k-dimensional random vectors. The component variables of a random vector. Definition of the joint law of a random vector:
the discrete case and the absolutely continuous case. The joint probability distribution function. Bivariate discrete finite
distributions: marginal and conditional distributions.
2. Discrete distributions.
General bivariate discrete distributions. Marginal distributions. Multivariate discrete distributions. Marginal distributions.
The multinomial distribution. Functions of a discrete random vector.
3. Continuous distributions.
General bivariate continuous distributions. Marginal distributions. Multivariate continuous distributions. Marginal
distributions. Functions of a continuous random vector.
4. Independence and conditional distributions.
Statistically independent random variables and joint distributions. Conditional distributions: discrete and continuous case.
5. Mathematical expectation and other numerical characteristics.
Expectation of a function of a random vector. The moment generation function. Covariance and correlation coefficient.
The variance-covariance matrix. Conditional expectation. Conditional variance. The double expectation theorem.
6. The multidimensional normal distribution.
The bidimensional normal distribution. The multidimensional normal distribution. Distributions related to the normal
distribution: chi-square distributions, Student t distributions and Fisher-Snedecor F distributions. Student theorem.
Cochran theorem.
Unless the requirements enforced by the health authorities demand a prioritization or reduction of these contents.
Methodology
Teaching methodology is based on the following activities and material:
- Theory lessons.
- Practical lessons about problems and exercises.
- Computer laboratory practical sessions (with Maxima and R).
- Personal work dossier (DTP).
- Theory and problem textbooks.
- Study and personal work weekly guides (GETPS).
- Course workspace on the UAB Virtual Campus Moodle.
The proposed teaching methodology may experience some modifications depending
on the restrictions to face-to-face activities enforced by health authorities.
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.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Theory lessons | 28 | 1.12 | 3, 2, 1, 6, 5, 4 |
| Type: Supervised | |||
| Computer laboratory practical sessions (with Maxima and R) | 14 | 0.56 | 3, 2, 1, 6, 5, 4 |
| Practical lessons about problems and exercises | 14 | 0.56 | 3, 2, 1, 6, 5, 4 |
| Type: Autonomous | |||
| Personal work dossier (exercices and theory) | 22 | 0.88 | 3, 2, 1, 6, 5, 4 |
Assessment
Continued evaluation:
Continued evaluation students can obtain up to 40% of the total score with personal work, done along the course:
personal work dossier (DTP), computer lab work (PRC) and classroom problems (EA), if delivered within the term.
The remaining score is covered by two (independent and equal weight) partial examinations, EP1 and EP2, both with a
second-chance examination, EF1 and EF2.
To pass the course, is required to obtain a minimum of 30% in each of both partial examinations, as well as a minimum of 50%
of the total score.
Denote DTP, EA, PRC, EP1, EP2, EF1 and EF2 the points (over 10) obtained on each of these evaluation items. Then the final
global punctuation QF (over 10) is calculated by means of the next formula:
QF = TC + 0.05 (10-TC-TC1) [max(EP1 , EF1) + max(EP2 , EF2)]
where TC = 0.2 DTP+0.1 EA+0.1 PRC and TC1 = max(0 , 1-0.2 DTP)+(1-0.1 EA) .
The minimum condition on partial examinations is: min{max(EP1 , EF1) , max(EP2 , EF2)} >=3
If this condition is not satisfied then the final global punctuation is min(QF , 4.5).
Single evaluation:
Single evaluation students will have a final exam and a recovery.
The final exam will be held on a single day and will have two parts, one in the morning and the other in the afternoon, with a maximum duration of 3 hours each.
The content of the exam in the morning will be the same as that of the EP1 exam (partial 1 of the continuous evaluation). We will denote AU1 the qualification of this exam, overf 10.
The content of the afternoon exam will be the same as that of the EP2 exam (partial 2 of the continuous evaluation). We will denote AU2 the qualification of this exam, over 10.
If the condition min(AU1, AU2) >= 3.5 is met, the final grade is calculated as QFU = (AU1 + AU2)/2, and the course is passed if QFU >= 5.
Otherwise, you have to take the recovery exam.
The recovery exam will be held on a single day and will have two parts, one in the morning and the other in the afternoon, with a maximum duration of 3 hours each.
The content of the exam in the morning will be the same as that of the EP1 exam (part 1 of the continuous evaluation). We will denote AUR1 the qualification of this exam, out of 10.
The content of the exam in the morning afternoon will be the same as that of the EP2 exam (partial 2 of the continuous evaluation). We will denote AUR2 the qualification of this exam, out of 10.
The minimum condition in this case is: min{max(AU1, AUR1), max(AU2, AUR2)} >= 3.5, and the final grade is calculated as
QFUR = 0.7 ([max (AU1, AUR1) + max (AU2, AUR2)]/2) + 0.3 QFU
If the minimum condition is not met, the final overall score is min(QFUR, 4.5).
Student’s assessment may experience somemodifications depending on the restrictions
to face-to-face activities enforced by health authorities.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Classroom problems (EA) | 10% | 14 | 0.56 | 3, 2, 1, 6, 5, 4 |
| Computer lab work (with Maxima and R) | 0% to 10% | 14 | 0.56 | 3, 2, 1, 6, 5, 4 |
| Final examinations (EF1, EF2) | 0% to 50% | 0 | 0 | 3, 2, 1, 6, 5, 4 |
| Partial examinations (EP1, EP2) | 20% to 40% (each one) | 0 | 0 | 3, 2, 1, 6, 5, 4 |
| Personal work dossier of exercices and theory (DTP) | 10% to 20% | 44 | 1.76 | 3, 2, 1, 6, 5, 4 |
Bibliography
J.E. Marsden & J. Tromba: Calculo Vectorial (Addison-Wesley).
M. de Groot: Probabilidad y Estadística (Addison-Wesley).
D. Peña: Fundamentos de Estadística (Alianza Editorial).(*)
D. Peña: Análisis de datos multivarianters (McGraw-Hill).(*)
J.G. Kalbfleisch: Probabilidad e Inferencia Estadística (Vol. 1) (AC).
V. Zaiats; M.L. Calle; R. Presas: Probabilitat i Estadística. Exercicis I. U.A.B. (Materials, 107).(*)
(*) most relevant bibliography.
Software
- R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
- wxMaxima (C) 2004-2018 Andrej Vodopivec