27012 - Introduction to Probability and Statistics
Basic Education
5.1. Methodological overview
The learning process will be mainly based on:
-Lectures on the theoretical topics listed in Section 5.3. The explanations will be illustrated by means of a variety of examples and real problems, trying to motivate the student`s participation. Computer presentations will be used. Such lectures will represent, at most, the 50% of the learning activities.
-Practical classes in which different exercises and questions will be solved in detail. The students will have in advance the whole collection of such exercises, in order to facilitate their homework. These practical sessions will represent, at least, the 40% of the learning activities.
-Practical sessions in computer labs, where the students will learn to use computer tools in order to solve exercises from a numerical viewpoint. They will represent, at least, the 10% of the learning activities.
-Individual tutorial sessions to discuss issues concerning the difficulties in the learning process, to correct the way of working, to monitorize the practical work assigned to each student,....
General information about the program, theoretical notes, collection of exercises, complementary material, references,...will be at the student`s disposal in the Moodle page of the Zaragoza University.
5.2. Learning tasks
In accordance with Section 5.2, the main learning activities will be the following:
-Two hours per week of theoretical lectures, addressed to all of the students in the course.
-Two hours per week of practical sessions, including those in computer labs. To carry out these activities, the students will be splitted into two groups.
-Other learning activities: individual tutorials, personal study and homework, works in groups,...
5.3. Syllabus
1. DATA ANALYSIS
1.1 Introduction: population and sample.
1.2 Relative frequencies and graphic representations.
1.3 Mean and standard deviation. Median and quantiles. Symmetry and kurtosis.
1.4 Outliers. Transformation of variables.
1.5 Two-dimensional data: joint, marginal and conditional distributions.
1.6 Moments. Covariance matrix and Pearson`s correlation coefficient.
1.7 The simplest linear model. Linear regression. Residuals analysis.
2. INTRODUCTION TO PROBABILITY
2.1 Sample space, events and algebras of events.
2.2 Axioms of probability. Consequences.
2.3 Classical probability. Combinatorics.
2.4 Finite, discrete, and geometric models. Examples.
2.5 Conditional probability and independence.
2.6 Total probability formula. Bayes formula.
3. DISCRETE RANDOM VARIABLES
3.1 Introductory examples. Probability laws and distribution functions.
3.2 The most usual distributions: uniform, Bernoulli and binomial, hypergeometric, geometric, negative binomial, and Poisson distributions.
3.3 Mathematical expectations. The expectation of a function of a discrete random variable.
3.4 Moments and central moments. Computations.
3.5 Moments and Chebyshev`s inequality.
3.6 Approximations: from the hypergeometric to the binomial, and from the binomial to the Poisson distributions.
4. ABSOLUTELY CONTINUOUS RANDOM VARIABLES
4.1 Introduction. Probability densities.
4.2 Distribution functions. Properties.
4.3 The most usual distributions: uniform, triangular, exponential, gamma, beta, Pareto, Cauchy, and normal distributions.
4.4 Transformations of absolutely continuous random variables. Change of variables.
4.5 Moments and central moments. Computations.
4.6 Moments and Chebyshev`s inequality.
4.7 The normal distribution: specific analysis and perspectives.
4.8 General random variables. Mixed random variables.