22.38 | Fall 2005 | Graduate

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

## Lecture Notes

LEC # TOPICS LECTURE NOTES
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

Risk-Informed Operational Decision Management (RIODM):

1. Risk, Event Trees and Fault Trees (PDF)
2. Reliability and Availability (PDF)

Structure Functions (PDF)

Valve Test Example (PDF)

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

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

Basic Probabilistic Concepts (PDF)

Convergence of Binomial and Normal Distributions for Large Numbers of Trials (PDF)

Convergence of Binomial and Poisson Distributions in Limiting Case of n Large, p<<1 (PDF)

Plane Crash Example (PDF)

V. Multivariate Distributions
9-10

V.1 Joint and Conditional Distribution Functions

V.2 Moments

V.3 The Multivariate Normal and Lognormal Distributions

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

VII. Statistical Methods
13-14

VII.1 Student’s t-distribution

VII.2 Chi-Squared Distribution

VII.3 Hypothesis Testing

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

Failure, Repair, Maintenance (PDF)

Reliability and Availability (PDF)

Operational Availability (PDF)

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

Bayes’ Theorem (PDF)

Bayesian Inference (PDF)

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

PRA: An Historical Perspective (PDF - 1.8 MB) (Courtesy of Prof. George Apostolakis. Used with permission.)

PRA Structure and Results (PDF - 1.1 MB)

Uncertainty (PDF)

Types of Uncertainty (PDF)

Common Cause Failures 1 (PDF)

Common Cause Failures 2 (PDF)

PRA in Managing Operations (PDF)

Engineered Safety Features (PDF)

Containment (PDF)

## Course Info

Fall 2005
##### Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes