Probability And Its Applications To Reliability, Q

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Probability And Its Applications To Reliability, Quality Control, And Risk Assessment

Fall 2005

Fault tree for external tank and solid rocket booster.

A fault tree from the NASA Accident Analysis Team in Report of the Presidential Commission on the Space Shuttle Challenger Accident. (Courtesy of NASA.)

Course Highlights

This course features a complete set of homework assignments and extensive lecture notes.

Course Description

This course covers interpretations of the concept of probability. Topics include basic probability rules; random variables and distribution functions; functions of random variables; and applications to quality control and the reliability assessment of mechanical/electrical components, as well as simple structures and redundant systems. The course also considers elements of statistics; Bayesian methods in engineering; methods for reliability and risk assessment of complex systems (event-tree and fault-tree analysis, common-cause failures, human reliability models); uncertainty propagation in complex systems (Monte Carlo methods, Latin Hypercube Sampling); and an introduction to Markov models. Examples and applications are drawn from nuclear and other industries, waste repositories, and mechanical systems.

Syllabus

Material Orientation

Probabilistic risk and reliability analyses provide methods for integrating the contribution of all contributors to ultimate system performance. They also permit the effects of uncertainties to be reflected directly in the understanding of system outcomes. In order to do this correctly, a good understanding of probabilistic concepts and methods is essential. This subject covers all of these topics.

Textbooks

Ang, A. H-S., and W. H. Tang. Probability Concepts in Engineering Planning and Design, vol. 1, Basic Principles. New York, NY: John Wiley & Sons, 1975. ISBN: 9780471032007.

Homework

Homework is assigned approximately weekly, due two class sessions after the date of assignment.

Grading

Course grading.
ACTIVITIES	PERCENTAGES
Homework	20%
Exam 1	20%
Exam 2	20%
Final Exam	40%

Calendar

Course calendar.
LEC #	TOPICS	KEY DATES
I. The Logic of Certainty
1-2	I.1 Events and Boolean Operations I.2 Event Sequence Identification (Failure Modes and Effects Analysis; Hazard and Operability Analysis; Fault Tree Analysis; Event Tree Analysis) I.3 Coherent Structure Functions I.4 Minimal Cut (Path) Sets
II. Probability
3-4	II.1 Definitions and Interpretations (Axiomatic; Subjectivistic; Frequentistic) II.2 Basic Rules II.3 Theorem of Total Probability II.4 Bayes' Theorem	Problem set 1 due
III. Random Variables and Distribution Functions
5-6	III.1 Discrete and Continuous Random Variables III.2 Cumulative Distribution Functions III.3 Probability Mass and Density Functions III.4 Moments III.5 Failure Models and Reliability III.6 Failure Rates
IV. Useful Probability Distributions
7-8	IV.1 Bernoulli Trials and the Binomial Distribution IV.2 The Poisson Distribution IV.3 The Exponential Distribution IV.4 The Normal and Lognormal Distributions IV.5 The Concept of Correlation	Problem set 2 due
V. Multivariate Distributions
9-10	V.1 Joint and Conditional Distribution Functions V.2 Moments V.3 The Multivariate Normal and Lognormal Distributions	Problem set 3 due
	Exam 1
VI. Functions of Random Variables
11-12	VI.1 Single Random Variable VI.2 Multiple Random Variables VI.3 Moments of Functions of Random Variables VI.4 Approximate Evaluation of the Mean and Variance of a Function VI.5 Analytical Results for the Normal and Lognormal Distributions	Problem set 4 due
VII. Statistical Methods
13-14	VII.1 Student's t-distribution VII.2 Chi-Squared Distribution VII.3 Hypothesis Testing	Problem set 5 due
VIII. Elements of Statistics
15	VIII.1 Random Samples VIII.2 Method of Moments VIII.3 Method of Maximum Likelihood VIII.4 Probability Plotting
IX. Applications to Reliability
16	IX.1 Simple Logical Configurations (Series; Parallel; Standby Redundancy) IX.2 Complex Systems IX.3 Stress-Strength Interference Theory IX.4 Modeling of Loads and Strength IX.5 Reliability-Based Design IX.6 Elementary Markov Models	Problem set 6 due
X. Bayesian Statistics
17	X.1 Bayes' Theorem and Inference X.2 Conjugate Families of Distributions X.3 Comparison with Frequentist Statistics X.4 Elicitation and Utilization of Expert Opinions
	Exam 2
XI. Monte Carlo Simulation
18	XI.1 The Concept of Simulation XI.2 Generation of Random Numbers XI.3 Generation of Jointly Distributed Random Numbers XI.4 Latin Hypercube Sampling XI.5 Examples from Risk and Reliability Assessment	Problem set 7 due
XII. Probabilistic Risk Assessment of Complex Systems
19-23	XII.1 Risk Curves and Accident Scenario Identification XII.2 Event-Tree and Fault-Tree Analysis XII.3 Unavailability Theory of Repairable and Periodically Tested Systems XII.4 Dependent (Common-Cause) Failures XII.5 Human Reliability Models XII.6 Component Importance XII.7 Examples from Risk Assessments for Nuclear Reactors, Chemical Process Systems, and Waste Repositories	Problem set 8 due Problem set 9 due Problem set 10 due
	Final Exam

Readings

Students are assigned readings from the textbooks for each class lecture topic, as listed in the following table. Additional reference books are also listed after the table.

Textbooks

Ang, A. H-S., and W. H. Tang. Probability Concepts in Engineering Planning and Design, vol. 1, Basic Principles. New York, NY: John Wiley & Sons, 1975. ISBN: 9780471032007.

Rausand, M., and A. Hoyland. System Reliability Theory: Models, Statistical Methods, and Applications. 2nd ed. New York, NY: John Wiley & Sons, 2003. ISBN: 9780471471332.

Course readings.
LEC #	TOPICS	READINGS
I. The Logic of Certainty
1-2	I.1 Events and Boolean Operations I.2 Event Sequence Identification (Failure Modes and Effects Analysis; Hazard and Operability Analysis; Fault Tree Analysis; Event Tree Analysis) I.3 Coherent Structure Functions I.4 Minimal Cut (Path) Sets	Ang and Tang: Chapters 1 and 2. Rausand and Hoyland: Sections 3.9-3.11.
II. Probability
3-4	II.1 Definitions and Interpretations (Axiomatic; Subjectivistic; Frequentistic) II.2 Basic Rules II.3 Theorem of Total Probability II.4 Bayes' Theorem
III. Random Variables and Distribution Functions
5-6	III.1 Discrete and Continuous Random Variables III.2 Cumulative Distribution Functions III.3 Probability Mass and Density Functions III.4 Moments III.5 Failure Models and Reliability III.6 Failure Rates	Ang and Tang: Sections 3.1 and 3.2. Rausand and Hoyland: Chapters 2, 4 and 6. Henley, E. J., and H. Kumamoto. "Risk Study Comparison." Table 1.4 in Probabilistic Risk Assessment. Piscataway, NJ: IEEE Press, 1991, pp. I.6. ISBN: 9780879422905.
IV. Useful Probability Distributions
7-8	IV.1 Bernoulli Trials and the Binomial Distribution IV.2 The Poisson Distribution IV.3 The Exponential Distribution IV.4 The Normal and Lognormal Distributions IV.5 The Concept of Correlation
V. Multivariate Distributions
9-10	V.1 Joint and Conditional Distribution Functions V.2 Moments V.3 The Multivariate Normal and Lognormal Distributions	Ang and Tang: Section 3.3. Rausand and Hoyland: Chapter 7.
	Exam 1
VI. Functions of Random Variables
11-12	VI.1 Single Random Variable VI.2 Multiple Random Variables VI.3 Moments of Functions of Random Variables VI.4 Approximate Evaluation of the Mean and Variance of a Function VI.5 Analytical Results for the Normal and Lognormal Distributions	Ang and Tang: Chapter 4.
VII. Statistical Methods
13-14	VII.1 Student's t-distribution VII.2 Chi-Squared Distribution VII.3 Hypothesis Testing	Ang and Tang: Chapters 5 and 6. Rausand and Hoyland: Chapters 11 and 14.
VIII. Elements of Statistics
15	VIII.1 Random Samples VIII.2 Method of Moments VIII.3 Method of Maximum Likelihood VIII.4 Probability Plotting
IX. Applications to Reliability
16	IX.1 Simple Logical Configurations (Series; Parallel; Standby Redundancy) IX.2 Complex Systems IX.3 Stress-Strength Interference Theory IX.4 Modeling of Loads and Strength IX.5 Reliability-Based Design IX.6 Elementary Markov Models	Ang and Tang: Chapter 9. Rausand and Hoyland: Chapters 8 and 9. Rausand, Marvin. "Reliability centered maintenance." Reliability Engineering and System Safety 60 (1998): 121-132. Vatn, J., P. Hokstad, and L. Bodsberg. "An overall model for maintenance optimization." Reliability Engineering and System Safety 51 (1996): 241-257.
X. Bayesian Statistics
17	X.1 Bayes' Theorem and Inference X.2 Conjugate Families of Distributions X.3 Comparison with Frequentist Statistics X.4 Elicitation and Utilization of Expert Opinions	Ang and Tang: Chapter 8. Rausand and Hoyland: Chapter 13. Huelsenbeck, J. P., et al. "Bayesian Inference of Phylogeny and Its Impact on Evolutionary Biology." Science 294, no. 5550 (December 14, 2001): 2310-2314.
	Exam 2
XI. Monte Carlo Simulation
18	XI.1 The Concept of Simulation XI.2 Generation of Random Numbers XI.3 Generation of Jointly Distributed Random Numbers XI.4 Latin Hypercube Sampling XI.5 Examples from Risk and Reliability Assessment
XII. Probabilistic Risk Assessment of Complex Systems
19-23	XII.1 Risk Curves and Accident Scenario Identification XII.2 Event-Tree and Fault-Tree Analysis XII.3 Unavailability Theory of Repairable and Periodically Tested Systems XII.4 Dependent (Common-Cause) Failures XII.5 Human Reliability Models XII.6 Component Importance XII.7 Examples from Risk Assessments for Nuclear Reactors, Chemical Process Systems, and Waste Repositories	Rausand and Hoyland: Chapters 5 and 10.
	Final Exam

Reference Books

In addition to the course textbooks, the following books may be useful.

Benjamin, Jack R., and C. Allin Cornell. Probability, Statistics, and Decision for Civil Engineers. 2nd ed. New York, NY: McGraw-Hill, 1963. ISBN: 9780070045583.

Center for Chemical Process Safety. Guidelines for Chemical Process Quantitative Risk Analysis. New York, NY: The American Institute of Chemical Engineers, 1989. ISBN: 9780816904020 .

DelMar, Donald, and George W. Sheldon. Introduction to Quality Control. St. Paul, MN: West Publishing Co., 1988. ISBN: 9780314684592.

Henley, Ernest J., and Hiromitsu Kumamoto. Probabilistic Risk Assessment: Reliability Engineering, Design, and Analysis. New York, NY: IEEE Press, 1991. ISBN: 9780879422905.

Juran, Joseph M., and Frank M. Gryna. Quality Planning and Analysis: From Product Development Through Use. 2nd ed. New York, NY: McGraw-Hill, 1980. ISBN: 9780070331785.

Papoulis, Athanasios. Probability and Statistics. Englewood Cliffs, NJ: Prentice Hall, 1990. ISBN: 9780137116980.

Rao, Singiresu S. Reliability-Based Design. New York, NY: McGraw-Hill, 1992. ISBN: 9780070511927.