Statistics and Probability Theory Lecture Notes Prof. Dr. M. H. Faber SS 2007 Copyright © 2007 Michael H. Faber PREAMBLE Introduction The present script serves as study guidance for the students taking the course on Statistics and Probability Theory at the summer semester at ETH. The present script provides information concerning the: • Aim of the course. • Structure and organisation of the course. • Educational support material for the course. • Mode of tests and exam. • Lecture notes for each of the 13 lectures with bibliography and index. Information about the contents of the course and the organization of the course is also available on http://www.ibk.ethz.ch/fa/. Aim of the course The aim of the present course is to provide to the students the basic skills and tools of statistics and probability. Emphasis is directed on the application and the reasoning behind the application of these skills and tools for the purpose of enhancing engineering decision making. It is expected that the students have only little or no prior knowledge on the subject of statistics and probability. The purpose of the present course is thus to ensure that the students will acquire during the course the required theoretical basis and technical skills such as to feel comfortable with the theory of basic statistics and probability. Moreover, in the present course as opposed to many standard courses on the same subject, the perspective is to focus on the use of the theory for the purpose of engineering model building and decision making. The course is subdivided into the following seven modules, each consisting of one or more lectures (see also Figure 1): • Module A - Engineering decisions under uncertainty Module B - Basic probability theory Module C - Descriptive statistics Module D - Uncertainty modelling Module E – Estimation and model building Module F – Methods of structural reliability Module G – Bayesian decision analysis i Module A Engineering decisions under uncertainty Module B - Basic probability theory Module C - Descriptive statistics 10 8 6 E1 E2 4 Number of observations 2 0 30-35 40-45 50-65 60-65 70-75 80-85 90-95 E E 100 105 110-115 1 2 Number of cars [102] Module E - Estimation and model building Module D - Uncertainty modeling Module F - Methods of structural reliability Module G - Bayesian decision analysis Figure 1: Illustration of the modules of the course and their didactical roles. The didactical logic behind the presentation of the material in the course is first to provide a motivation for the application of statistics and probability as a basis for developing engineering models and for risk based decision making (Module A, see Figure 1). Thereafter, a basic introduction on the theory of probability (Module B) is provided. This is meant as a “brush up” of the knowledge already acquired by the students during high-school. Subsequently in Module C a selection of tools is provided, which enables engineers to assess and communicate data in a condensed form, namely the descriptive statistics. In Module D an introduction to uncertainty is provided together with a description of the various building stones required to represent uncertainties in engineering modelling in terms of random variables and processes. In Module E the main focus is directed on the aspects of postulating models, assessing model parameters and verifying models. Subsequently in Module F it is shown how, on the basis of formulated probabilistic models of uncertain variables, probabilities of events of significance for engineering decision making may be assessed. Finally in Module G it is shown how the engineering models of uncertainties and their probabilistic descriptions can be utilized in a systematic framework for engineering decision making. It is believed that students having completed the present course will be able to: • assess data based on observations and/or experiment results and present these in a standardized and unambiguous form, ii • formulate and validate simple engineering models with due consideration of the associated uncertainties due to lack of knowledge and data as well as natural inherent variability, • perform simple probability assessments such as to evaluate the probability of the appropriate performance of engineering activities, • formulate and solve simple risk based decision problems. Structure and organization of the course With the aim of supporting the students in learning the specified curriculum, the course is built up by four main components, namely lectures, tutorials, assessments and self study: • 13 weekly lectures of each two sessions of 45 minutes • 11 weekly exercise tutorials of each two sessions of 45 minutes • 2 assessments of each 90 minutes • Self study estimated to 4 times by 45 minutes per week This scheme is in accordance with the presently considered best ETH practice which assumes that the total efforts required to complete the course correspond to; lectures + exercise tutorials+assessment = 50%, self study = 50%. Lectures: The lectures are targeted at providing the students with the most important aspects of the theoretical and methodical material which may also be found in the present lecture notes. However, the lectures will also focus on the philosophical background for the development and use of the theoretical background and are thus to be understood as partly complementary to the material of the lecture notes. It is assumed and strongly suggested that the students study the lecture notes and become familiar with this. Exercise Tutorials: The exercise tutorials serve as a means of learning how to apply the theories and methodologies presented in the lectures and in the script. It is expected that the student will actively engage themselves in the tutorial work. During the tutorials, provided that there is sufficient time, the students may also use the possibility to ask about any problems related to the solutions of the solved exercises which are provided on the home page. The students, furthermore, have the possibility to consult the teaching staff at defined office hours in regard to the contents of the lectures and the exercise tutorials. In order to enable the complete clarification of problems and questions regarding the teaching material, it is advisable to contact personally the teaching assistants, rather than making requests through E- mails. iii For the purpose of supporting the students in their self evaluation, solved exercises, including previous exam exercises are made available on the home page http://www.ibk.ethz.ch/fa/. In addition, in the lecture notes at the end of each chapter, there is a set of small principal exercises which the students can use to check and practice their knowledge and skills. During the first exercise tutorial the students will be sub-divided into groups which will have to present the solution of a representative exercise during the course. Each individual exercise tutorial includes the following activities: • Presentation of 2 or more new exercises (corresponding to the subjects presented in the last lecture) in steps that will enable their solution by the students. • Presentation by the teaching assistants of the solutions of 1 or more exercises of those presented in the previous exercise tutorial. • Presentation by one group of students of the solution of one of the exercises presented in the previous exercise tutorial. Student colleagues and teaching assistants may ask questions for clarification during the presentation. Educational support material for the course The course is supported by the present script which provides the theory being taught during the lectures. All material for the course will be made available partly prior to the start of the course and partly during the course on the home page http://www.ibk.ethz.ch/fa/. The course material contains besides the lecture notes also the Power Point presentations used for the lectures as well the solved exercises for each exercise tutorial. The lecture notes will be made available on the home page prior to the start of the course. The Power Point files of both lecture and exercise tutorials will be uploaded on the course’s web page the latest one day before the respective class. All solutions to the exercises, except the exercises for which the solutions will be presented by the students during the exercise tutorials will be made available on the home page prior to the start of the course. The solutions of the latter mentioned exercises will appear immediately after the presentations. The Power Point presentations are only meant as a support for the lectures and can be used only as a support for the learning and preparation before each lecture. It is important to note that it is expected that the students read all the material contained in the lecture notes from lecture to lecture. Reading the Power Point presentations is not a substitute for reading the lecture notes which in many cases provide more detailed information. Mode of Assessment (Tests and Exams) The course performance comprises of a joint evaluation of: iv • The results of the two assessments, one midterm (03.05.07) and the other one towards the end of the course (14.06.07) • The final exam which will take place in the autumn (close or within October) as a part of the “Basisprüfung” (to be announced by the ETH Prüfungsplan). The two assessments during the course have equal weight and must be attended by all students. In case documentation from a medical doctor or a military superior confirming illness or military duties, is presented to Prof. Faber before or within one week after a test which is not attended, arrangements for a substitute examination can be made. In case that an assessment is not attended by a student and no documentation is presented, the assessment will automatically be marked with 1 (1 out of 6, in the formal ETH scale with 6 being the best mark).