Probability and Statistics II

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Credits
6
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
EIO;MAT
The objective of this subject is to introduce the students in the basic concepts of Markov chains and Poisson processes. Also to introduce them in the construction of a good model for analyzing real data with continuous, discrete and cathegorical responses, using linear and generalized linear models. It will be specially important that the student gets familiarized with the analysis of real data and can obtain interesting conclusions from the,m.

Teachers

Person in charge

  • Marta Pérez Casany ( )

Others

  • Víctor Peña Pizarro ( )

Weekly hours

Theory
2
Problems
0
Laboratory
2
Guided learning
0
Autonomous learning
6

Competences

Technical Competences

Technical competencies

  • CE3 - Analyze complex phenomena through probability and statistics, and propose models of these types in specific situations. Formulate and solve mathematical optimization problems.

Transversal Competences

Transversals

  • CT5 - Solvent use of information resources. Manage the acquisition, structuring, analysis and visualization of data and information in the field of specialty and critically evaluate the results of such management.
  • CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

Basic

  • CB1 - That students have demonstrated to possess and understand knowledge in an area of ??study that starts from the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that imply Knowledge from the vanguard of their field of study.
  • CB3 - That students have the ability to gather and interpret relevant data (usually within their area of ??study) to make judgments that include a reflection on relevant social, scientific or ethical issues.
  • CB5 - That the students have developed those learning skills necessary to undertake later studies with a high degree of autonomy

Generic Technical Competences

Generic

  • CG1 - To design computer systems that integrate data of provenances and very diverse forms, create with them mathematical models, reason on these models and act accordingly, learning from experience.
  • CG2 - Choose and apply the most appropriate methods and techniques to a problem defined by data that represents a challenge for its volume, speed, variety or heterogeneity, including computer, mathematical, statistical and signal processing methods.

Objectives

  1. To learn how to contruct statistical models in order to sinthesize information, explain a response variable as a function of some explanatory variables, and do forecasting.
    Related competences: CE3, CT5, CG2, CB3, CB5,
  2. To understand the basic concepts and the philosophy behind Bayesian statistics.
    Related competences: CE3, CT6, CB1, CB5,
  3. To learn software statistics and how to use it to analyze real data
    Related competences: CE3, CT5, CT6, CG2, CB5,
  4. To learn model validation techniques.
    Related competences: CE3, CT5, CT6, CG2, CB3, CB5,
  5. To learn how to do a report containign the results of a data analysis
    Related competences: CT5, CG2,
  6. To understand the difference between the Bayesian and frequentist statistics
    Related competences: CE3, CT5, CG2, CB5,
  7. To know which is the most suitable modalization technique for each problem.
    Related competences: CE3, CG2, CB1, CB3, CB5,
  8. To learn how to interpret the results of a fitted model
    Related competences: CE3, CT6, CG2, CB1, CB3,
  9. To understand the concept of crossvalidation and the ones of overfitting and underfitting
    Related competences: CE3, CG2, CB1, CB3, CB5,
  10. To use the model fitted for predictions
    Related competences: CE3, CT5, CG2, CB1, CB3, CB5,
  11. To understand the difference between parameter and parameter estimation, to solve inference problems in linear and generalized linear models.
    Related competences: CE3, CG2, CB1, CB3, CB5,
  12. To learn how to include cathegorical variables in linear and generalized linear models.
    Related competences: CE3, CG2, CB1, CB3, CB5,
  13. To analyze with critical sense, data and topics relevant for the society
    Related competences: CT6, CG1, CB3, CB5,
  14. To perform estimation usign confidence intervals
    Related competences: CT5, CB3, CB5,
  15. To understand the importance of the hypothesis testing. To know how to perform the classical hypothesis tests and to know techniques to face new hypothesis test that can appear doing research.
    Related competences: CE3, CT6, CB3, CB5,

Contents

  1. Distributions related to the Normal distribution. Confidence Interval Estimation.
    Distribucions Chiquadrat, t-d'Student i F-Fisher-Snedecor. Definició d'intèrval de confiança. IC per un valor esperat, per una variància, per una probabilitat i per la diferència de dos valors esperats i dues probabilitats. Quantitats pivotals.
  2. Hypothesis testing
    Conceptes generals en l'entorn dels test d'hipòtesis. Comparcions d'una esperança i una variància amb un valor concret. Comparació de dos valors esperats, comparació de dues variàncies. Comparació d'una probabilitat amb un valor concret. Comparació de dues probabilitats.
  3. Linear model
    Linear model definition. Parameter estimation. Anova Table and goodness-of-fit measures. Inference. Prediction. Model validation. Model selection. Model interpretation. Bias, colliniarity. causality. Use of cathegorical explanatory variables.
  4. Generalized linear model
    Definition of generalized linear model. Models for binary response. Parameter estimation. Inference. Model validation. Model selection. prediction. Model interpretation.
  5. Introduction to Bayessian statistics.
    Bayes theorem. Bayessian model. Predictive distribution a priori and a posteriori. Selection of the a priori distribution.

Activities

Activity Evaluation act


Linear models

Linear model definition. Estimation and inference in a linear model. Model validation. Model selection. Model interpretation. Linear models with categorical variables. Non linear models with gaussian response.
Objectives: 1 3 4 5 7 8 10 11 12 13
Contents:
Theory
12h
Problems
0h
Laboratory
12h
Guided learning
0h
Autonomous learning
35h

Generalized linear model.

Generalized linear model definition. Models for binary response. Estimation and inference in a generalized linear model. Prediction, interpretation and model selection.
Objectives: 1 3 4 5 7 8 9 10 11 12 13
Contents:
Theory
6h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
15h

Bayesian statistics

Statistical model. Inference based on liekelihood. Bayesian model. "A posteriori" distribution. Predictive distribution "a priori" and "a posteriori". How to select the "a priori distribution". Bayesian inference. Model validation.
Objectives: 2 6
Theory
2h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
10h

Distributions related to the Normal distribution. Confidence interval estimation

The distributions chi-square, Student-t and Fisher are defined. The concept of Confidence interval and pivotal quantity are introduced. The most important and useful confidence intervals are computed.
Objectives: 13 14
Contents:
Theory
4h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
15h

Hypothesis test

The basic concepts related to hypothesis test are introduced. The hypothesis for comparing one mean and one variance to a given value, to compare two means, two variances and to probabilities are shown.
Objectives: 5 13 15
Contents:
Theory
6h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
15h

Teaching methodology

One half of the sessions will consist on the exposition of new concepts and contents. The other half will be devoted to solve practical exercises. At the end of each practical session, some exercises will be proposed to the students in order that they can work i autonomously.

Evaluation methodology

There will be a partial exam and a final exam, as well as exercises of data analysis assigned during the course.

The partical exam will correspond to the confidence intervals and hypothesis tests.

The final exam will correspond to the rest of the subject contents.

The course mark will be the sample mean of the exercices realized during the course.

The final mark will be computed as:

Subject Mark = 0.25 * Cours+ 0.25 * Partial+ 0.5 *FinalExam

In the case of the students that go to the reevaluation, the final mark will be computed like this:

Subject Mark=max(Reevaluation, 0,25Cours+0,75Reevaluation)

Bibliography

Basic:

Previous capacities

To follow this subject, the student needs to have a good understanding of the previous subjects entitled: PiE1 and Calcul.