# Probability and Statistics I

## You are here

Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT;EIO
The subject introduces the basic concepts of probability and statistics that are necessary for the development of applications in Data Science and Engineering. It provides the conceptual framework and basic instrumental skills in probability theory and stochastical processes, as well as an introduction to statistical inference.

## Teachers

### Person in charge

• Guillem Perarnau Llobet ( )

### Others

• Anna De Mier Vinué ( )
• Cristian Pachón Garcia ( )

## Weekly hours

Theory
3
Problems
1
Laboratory
1
Guided learning
0
Autonomous learning
7.5

## 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 [Avaluable] - 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 - 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. At the end of the course, students will know the definition of probability and their properties, and will apply them to solve probability calculation problems.
Related competences: CE3, CG1, CB5,
2. At the end of the course students will know how to use the concept of random variable to formalize and solve probability calculation problems.
Related competences: CE3, CG1, CB5,
3. At the end of the course students will know how to simulate complex random phenomena with the computer and deduce approximate values of amounts of interest (probabilities, characteristics of random variables) that are difficult to calculate analytically.
Related competences: CE3, CT5, CT6, CG1, CG2, CB1, CB3, CB5,
4. At the end of the course students will know the most common probabilistic distributions and will be able to recognize situations where they are used to model real phenomena.
Related competences: CE3, CG1, CB5,
5. At the end of the course, students will know how to calculate distributions and expected expectations and use them in prediction.
Related competences: CE3, CT6, CG1, CB5,
6. At the end of the course, students will know whether two random variables are independent, and if they are not, they will be able to measure the linear correlation coefficient.
Related competences: CE3, CT6, CG1, CB5,
7. At the end of the course, students will know the Law of the Great Names and the Central Limit Theorem.
Related competences: CE3, CG1, CB1, CB5,
8. At the end of the course, the students will understand stochastic processes and will know how to model random-flavoured problems using Markov chains.
Related competences: CE3, CT6, CG1, CG2, CB1,
9. At the end of the course students will know the basic tools of descriptive statistics and will know how to apply them.
Related competences: CE3, CT5, CG1, CG2, CB3, CB5,
10. At the end of the course students will know the concepts of population, sample, parameter and estimator, and know the basic properties.
Related competences: CE3, CT6, CG1, CB5,
11. At the end of the course, students will know the basics of timely estimation and will know how to calculate them in real situations
Related competences: CE3, CT5, CT6, CG1, CB3, CB5,

## Contents

1. Probability spaces and random variables
Random experiences. Algebra of events. Probability space. Conditional probability. Independence of events. Bayes' Theorem. Simulation of random experiments.
2. Random variables
Definition of random variable. Probability distribution function. Discrete random variables (probability function) and continuous variables (probability density function). Expectation and moments. Models of usual distributions. Simulation of random variables.
3. Random vectors
Multidimensional distributions. Independence. Conditioned distributions. Covariance and correlation. Expectation and covariance matrix. Conditioned expectation. Multinomial distribution. Multivariate normal distribution.
4. Sum of random variables
Distribution of the sum. Markov's, Chebyshev's and Chernoff's Inequalities. Law of Large Numbers. Central Limit Theorem.
5. Stochastic processes
Stochastic processes. Markov chains. Recurrence and transience. Ergodic theorem.
6. Population and sample
Random sample. Parametric statistical models. Parameters and estimators. Descriptive statistics.
7. Point estimation
Method of moments. Maximum likelihood. Properties of the estimators (bias, variance, mean square error, consistency).

## Activities

Activity Evaluation act

### Developing the Topic "Probability and random variables"

Developing the Topic "Probability and random variables"
Objectives: 1 3
Contents:
Theory
5h
Problems
2h
Laboratory
2h
Guided learning
0h
Autonomous learning
8h

### Developing the Chapter "Random variables"

Developing the Chapter "Random variables"
Objectives: 2 3 4 9
Theory
5h
Problems
2h
Laboratory
2h
Guided learning
0h
Autonomous learning
8h

### Developing the Chapter "Random vectors"

Developing the Chapter "Random vectors"
Objectives: 3 5 6 9
Contents:
Theory
7.5h
Problems
3h
Laboratory
2h
Guided learning
0h
Autonomous learning
6.3h

### Developing the Chapter "Sum of random variables"

Developing the Chapter "Sum of random variables"
Objectives: 3 5 7
Contents:
Theory
5h
Problems
1.5h
Laboratory
0h
Guided learning
0h
Autonomous learning
7.8h

### Developing the Chapter "Stochastic Processes"

Developing the Chapter "Stochastic Processes"
Objectives: 8
Contents:
Theory
8.5h
Problems
3h
Laboratory
3h
Guided learning
0h
Autonomous learning
11.2h

### Developing the Chapter "Population and sample"

Developing the Chapter "Population and sample"
Objectives: 3 10 9
Contents:
Theory
3.5h
Problems
2h
Laboratory
2h
Guided learning
0h
Autonomous learning
4.5h

### Developing the Chapter "Point estimation"

Developing the Chapter "Point estimation"
Objectives: 11
Contents:
Theory
5.5h
Problems
1.5h
Laboratory
1.5h
Guided learning
0h
Autonomous learning
6.7h

### Final examination

Final examination
Objectives: 1 2 4 5 6 7 10 11
Week: 15 (Outside class hours)
Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
30h

### Mid-term examination

Mid-term examination
Objectives: 1 2 4 5 6
Week: 8 (Outside class hours)
Theory
2h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
10h

### Mid-term lab examination

Mid-term lab examination
Objectives: 3 9
Week: 8 (Outside class hours)
Theory
0h
Problems
0h
Laboratory
1h
Guided learning
0h
Autonomous learning
10h

### Final lab examination

Final lab examination
Objectives: 3 9 11
Week: 15 (Outside class hours)
Theory
0h
Problems
0h
Laboratory
1.5h
Guided learning
0h
Autonomous learning
10h

## Teaching methodology

Theory:
Lectures develop the theory and include illustrative examples.

Problems:
The students have in advance the list of problems relevant to the topic being developed in theory. They had the opportunity to try to solve problems before the problems class. They require the teacher's help in the points where they have encountered difficulties. The teacher solves these questions in the blackboard and develops the full solution of some problems that he or she considers that are especially challenging.

Laboratory:
The teacher introduces the R language during the course, with special emphasis on random variables simulation tools, descriptive statistics and univariate statistical inference.

## Evaluation methodology

A midterm exam (EP) and a final exam (EF). The midterm exam will assess the first part of the course, and the final exam, the second one. Each of them has a part of theory and problems, and may contain a part of laboratory. Optionally, the day of the final exam it will be possible to resit the midterm exam (REP), if the exam is submitted, its mark will replace the mark of the midterm exam.

During the course short activities or assignments (ACP) will be proposed.

The final grade (NF) is computed as follows: if the resit is not submitted

NF = 0.45·EP +0.45·EF +0.10·ACP,

and if the resit exam is submitted

NF =0.45·REP +0.45·EF +0.10·ACP.

Only students with NF smaller than 5 can opt to re-evaluation. The re-evaluation exam grade (ER) replaces the 100% of the midterm and final exams grade. So the final grade after re-evaluation (NFreav) will be

NFreav = 0.90 * ER + 0.10*ACP.

In case that NFreav is smaller than 5, the final mark will be the maximum between NF and NFreav