ENGINEERING

UNCERTAINTY AND RISK ANALYSIS

A Balanced Approach to Probability, Statistics,

Stochastic Modeling, and Stochastic Differential Equations

Sergio E. Serrano, Ph.D.

1 ENGINEERING UNCERTAINTY ANALYSIS

1.1 PROBABILISTIC ANALYSIS OF ENGINEERING SYSTEMS

• The Engineering Method
• The Origins of Uncertainty
• Deterministic Versus Probabilistic Analysis of Engineering Systems

1.2 METHOD OF UNCERTAINTY ANALYSIS

QUESTIONS AND PROBLEMS

2 THE CONCEPT OF PROBABILITY

2.1. CHANCE EXPERIMENTS AND THEIR OUTCOMES

2.2 ESTIMATION OF THE NUMBER OF POSSIBLE OUTCOMES

• Simple Enumeration
• Permutations: Sampling in a Specific Order Without Replacement
• Combinations: Sampling Without a Specific Order and Without Replacement

2.3 SAMPLE SPACE AND EVENTS

2.4 QUANTITATIVE EVALUATION OF PROBABILITY

• The Frequency Definition of probability
• Fundamental Axioms of Probability
• Additional Probability Relationships

2.5 INDEPENDENCE AND CONDITIONING OF PROBABILISTIC EVENTS

• Statistical Independence
• The Total Probability and Bayes' Theorems

PROBLEMS

3 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3.1 RANDOM VARIABLE: A FUNCTION OF PROBABILITY

3.2 DISCRETE RANDOM VARIABLES

• The Probability Mass Function
• The Cumulative Distribution Function
• Expected Value of a Random Variable
• The Variance of a Random Variable
• The Binomial Random Variable
• The Geometric Random Variable
• The Pascal Random Variable
• The Hypergeometric Random Variable
• The Poisson Random Variable
• Approximation to the Poisson Random Variable

3.3 CONTINUOUS RANDOM VARIABLES

• The Probability Density Function
• The Cumulative Distribution Function
• Mean, Variance, and other Moments of Continuous Random Variables
• The Exponential Random Variable
• The Uniform Random Variable
• The Gaussian Random Variable
• The Lognormal Random Variable
• The Gamma Random Variable
• Other Probability Distributions

PROBLEMS

4 SIMULATION OF RANDOM SYSTEMS

4.1 DERIVATION OF A SYSTEM OUTPUT DENSITY FUNCTION

4.2 MONTE CARLO SIMULATION

• Generation of Random Numbers from a Specified Density Function
• Analytical Decomposition of the Transformation Equation
• Numerical Solution of the Transformation Equation
• The Heart of Monte Carlo Simulations: Generation of Uniform Random Numbers
• Generation of Gaussian Random Numbers
• Application of the Transformation Equation: Generation of Random Numbers of Other Probability Distributions

4.3 SIMULATION OF SYSTEMS WITH SEVERAL RANDOM VARIABLES

• System Sensitivity to Uncertainty in One or More Variables
• Systems with Several Independent Random Variables

4.4 ANALYTICAL DERIVATION OF SECOND-ORDER STATISTICS

PROBLEMS

5 SYSTEMS WITH JOINTLY-DISTRIBUTED RANDOM VARIABLES

5.1 TWO DISCRETE RANDOM VARIABLES

• The Joint Probability Mass Function
• The Joint Cumulative Distribution Function
• Marginal Distributions
• Conditional Joint Functions

5.2 TWO CONTINUOUS RANDOM VARIABLES

• The Joint Probability Density Function
• The Joint Cumulative Distribution Function
• The Marginal Probability Density Functions
• Conditional Joint Functions
• Statistically Independent Random Variables

5.3 SPECIAL MOMENTS OF TWO RANDOM VARIABLES

• The Covariance of Two Random Variables
• The Correlation Coefficient of Two Random Variables

5.4 THE BIVARIATE GAUSSIAN DENSITY FUNCTION

5.5 SYSTEM S FORCED BY JOINTLY-DISTRIBUTED RANDOM VARIABLES

• Output of Sums of Random Variables: The Central Limit Theorem

PROBLEMS

6 ESTIMATION THEORY IN ENGINEERING

6.1 STATISTICS AND UNCERTAINTY ANALYSIS

• Population Parameters versus Sample Statistics
• Probability Distribution of the Sample Mean

6.2 POINT ESTIMATORS

• Estimation with the Method of Moments
• Estimation with the Method of Maximum Likelihood

6.3 INTERVAL ESTIMATORS

• Confidence Intervals
• Case 1: Confidence Interval for the Mean (Variance Known)
• Case 2: Confidence Interval for the Mean (Variance Unknown)
• Case 3: Confidence Interval for the Variance (Mean Known)
• Case 4: Confidence Interval for the Variance (Mean Unknown)

6.4 STATISTICAL TESTS

• Case 1: Test for the Population Mean (Variance Known)
• Case 2: Test for the Population Mean
• Case 3: Test for the Population Mean (Variance Unknown and N<60)
• Case 4: Test for the Population Variance
• Case 5: Test for the Population Variance ( N>100)

PROBLEMS

7 FITTING PROBABILITY MODELS TO DATA

7.1 EMPIRICAL DISTRIBUTIONS

• The Frequency Histogram from Observed Data
• The Expected Frequency Histogram from a Theoretical Distribution
• The Empirical Cumulative Distribution Function

7.2 STATISTICAL TESTS FOR GOODNESS OF FIT

• The Chi Squared Goodness of Fit Test

PROBLEMS

8 REGRESSION ANALYSIS

8.1 STATISTICAL MEASURES BETWEEN TWO RANDOM VARIABLES

• The Sample Covariance and Sample Correlation Coefficient

8.2 THE LEAST-SQUARES STRAIGHT LINE

• Fitting a Straight Line through the Scatter Diagram
• Non-Linear Curves Reducible to Straight Lines

8.3 CONFIDENCE INTERVALS OF THE REGRESSION MODEL

• Case 1: Confidence Interval of the Slope a
• Case 2: Confidence Interval of the Slope a
• Case 3: Confidence Interval of the Predicted Y
• Case 4: Confidence Interval of the Predicted Y
• Case 5: Confidence Interval of the Intercept b
• Case 6: Confidence Interval of the Intercept b

PROBLEMS

9 RELIABILITY OF ENGINEERING SYSTEMS

9.1 THE CONCEPT OF RELIABILITY

9.2 TIME RELIABILITY

9.3 RELIABILITY OF SYSTEMS

• Systems in Series
• Systems in Parallel
• Hybrid Systems

9.4 ENGINEERING MODELS OF FAILURE

• The Weibull Failure Model
• Fitting Data to a Weibull Failure Model

PROBLEMS

10 DESIGN OF ENGINEERING EXPERIMENTS

10.1 THE CONCEPT OF STATISTICAL EXPERIMENT DESIGN

10.2 ESTIMATING THE POPULATION MEAN FROM LIMITED SAMPLING

10.3 ESTIMATING THE REQUIRED NUMBER OF MEASUREMENTS

• Pre-Specified Variance
• Pre-Specified Margin of Error

PROBLEMS

11 EXPERIMENTS AND TESTS FOR TWO OR MORE POPULATIONS

11.1 COMPARISON OF PARAMETERS OF TWO POPULATIONS

• Case 1: Test for the Comparison of Means of Two Populations
• Case 2: Test for the Comparison of Means of Two Populations
• Case 3: Test for the Comparison of Means of Two Populations
• Case 4: Test for the Comparison of Means of Two Populations
• Case 5: Test for the Comparison of Means of Two Populations
• Case 6: Test for the Comparison of Variances of Two Normal Populations

11.2 COMPARISON OF MEANS OF TWO OR MORE POPULATIONS: SINGLE FACTOR ANALYSIS OF VARIANCE (ANOVA)

• Box Plots and the Logic Behind ANOVA
• Case 7: Test for the Comparison of Means of More than Two Normal Populations

PROBLEMS

12 STOCHASTIC PROCESSES

12.1 THE CONCEPT OF A STOCHASTIC PROCESS

12.2 FIRST AND SECOND-ORDER STATISTICS

• Density Function and Cumulative Distribution Function
• Mean, and Correlation Functions
• Stationarity
• The Correlogram
• Transformations of the Correlation Function: The Spectral Density

12.3 SOME THEORETICAL STOCHASTIC PROCESSES

• The Random Walk Process
• The Brownian Motion Process
• The White Gaussian Noise Process

12.4 TIME SERIES ANALYSIS

• Time Average Versus Ensemble Properties
• Deterministic Trend
• Periodicity
• Models for the Random Component

PROBLEMS

13 STOCHASTIC DIFFERENTIAL EQUATIONS

13.1. THE ORIGIN OF STOCHASTIC DIFFERENTIAL EQUATIONS

13.2 STOCHASTIC CONTINUITY, DIFFERENTIATION, AND INTEGRATION

• Mean Square Continuity
• Stochastic Differentiation
• Stochastic Integrals

13.3 SOLVING APPLIED STOCHASTIC DIFFERENTIAL EQUATIONS

• Differential Equations with Random Initial Conditions
• Differential Equations with Random Forcing Functions
• Solution of Random Equations with Decomposition
• Solving Non-Linear Differential Equations
• Differential Equations with Random Coefficients

PROBLEMS

14 CONCLUSION

APPENDIX A: TABLES

• Table A.1: Cumulative Areas under the Standard Normal Probability Density Function
• Table A.2: Abscissa Values Corresponding to Areas under the Student's t Density Function with m Degrees of Freedom
• Table A.3: Abscissa Values Corresponding to Areas under the Chi-Squared Density Function with m Degrees of Freedom
• Table A.4: Abscissa Values Corresponding to Areas under the F Density Function with and Degrees of Freedom

APPENDIX B: ANSWERS TO PROBLEMS

BIBLIOGRAPHY

INDEX

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