Engineering Uncertainty Analysis | p. 1 |

Probabilistic Analysis of Engineering Systems | p. 1 |

The Engineering Method | p. 1 |

The Origins of Uncertainty | p. 3 |

Deterministic Versus Probabilistic Analysis of Engineering Systems | p. 5 |

Method of Uncertainty Analysis | p. 12 |

Questions and Problems | p. 19 |

The Concept of Probability | p. 21 |

Chance Experiments and Their Outcomes | p. 21 |

Estimation of the Number of Possible Outcomes | p. 22 |

Simple Enumeration | p. 22 |

Permutations: Sampling in a Specific Order Without Replacement | p. 24 |

Combinations: Sampling Without a Specific Order and Without Replacement | p. 25 |

Sample Space and Events | p. 27 |

Quantitative Evaluation of Probability | p. 28 |

The Frequency Definition of probability | p. 28 |

Fundamental Axioms of Probability | p. 31 |

Additional Probability Relationships | p. 38 |

Independence and Conditioning of Probabilistic Events | p. 38 |

Statistical Independence | p. 40 |

The Total Probability and Bayes' Theorems | p. 51 |

Problems | p. 54 |

Random Variables and Probability Distributions | p. 57 |

Random Variable: A Function of Probability | p. 57 |

Discrete Random Variables | p. 59 |

The Probability Mass Function | p. 59 |

The Cumulative Distribution Function | p. 61 |

Expected Value of a Random Variable | p. 63 |

The Variance of a Random Variable | p. 65 |

The Binomial Random Variable | p. 69 |

The Geometric Random Variable | p. 73 |

The Pascal Random Variable | p. 74 |

The Hypergeometric Random Variable | p. 75 |

The Poisson Random Variable | p. 76 |

Approximation to the Poisson Random Variable | p. 79 |

Continuous Random Variables | p. 80 |

The Probability Density Function | p. 80 |

The Cumulative Distribution Function | p. 85 |

Mean, Variance, and other Moments of Continuous Random Variables | p. 87 |

The Exponential Random Variable | p. 91 |

The Uniform Random Variable | p. 93 |

The Gaussian Random Variable | p. 96 |

The Lognormal Random Variable | p. 101 |

The Gamma Random Variable | p. 103 |

Other Probability Distributions | p. 106 |

Problems | p. 107 |

Simulation of Random Systems | p. 111 |

Derivation of a System Output Density Function | p. 111 |

Monte Carlo Simulation | p. 116 |

Generation of Random Numbers from a Specified Density Function | p. 117 |

Analytical Decomposition of the Transformation Equation | p. 122 |

Numerical Solution of the Transformation Equation | p. 125 |

The Heart of Monte Carlo Simulations: Generation of Uniform Random Numbers | p. 128 |

Generation of Gaussian Random Numbers | p. 131 |

Application of the Transformation Equation: Generation of Random Numbers of Other Probability Distributions | p. 133 |

Simulation of Systems with Several Random Variables | p. 135 |

System Sensitivity to Uncertainty in One or More Variables | p. 135 |

Systems with Several Independent Random Variables | p. 136 |

Analytical Derivation of Second-Order Statistics | p. 141 |

Problems | p. 144 |

Systems with Jointly-Distributed Random Variables | p. 147 |

Two Discrete Random Variables | p. 147 |

The Joint Probability Mass Function | p. 147 |

The Joint Cumulative Distribution Function | p. 150 |

Marginal Distribution | p. 152 |

Conditional Joint Functions | p. 153 |

Two Continuous Random Variables | p. 155 |

The Joint Probability Density Function | p. 155 |

The Joint Cumulative Distribution Function | p. 157 |

The Marginal Probability Density Functions | p. 159 |

Conditional Joint Functions | p. 160 |

Statistically Independent Random Variables | p. 162 |

Special Moments of Two Random Variables | p. 162 |

The Covariance of Two Random Variables | p. 163 |

The Correlation Coefficient of Two Random Variables | p. 164 |

The Bivariate Gaussian Density Function | p. 166 |

System s Forced by Jointly-Distributed Random Variables | p. 168 |

Output of Sums of Random Variables: The Central Limit Theorem | p. 169 |

Problems | p. 174 |

Estimation Theory in Engineering | p. 177 |

Statistics and Uncertainty Analysis | p. 177 |

Population Parameters versus Sample Statistics | p. 178 |

Probability Distribution of the Sample Mean | p. 181 |

Point Estimators | p. 182 |

Estimation with the Method of Moments | p. 183 |

Estimation with the Method of Maximum Likelihood | p. 183 |

Interval Estimators | p. 186 |

Confidence Intervals | p. 186 |

Confidence Interval for the Mean (Variance Known) | p. 187 |

Confidence Interval for the Mean (Variance Unknown) | p. 191 |

Confidence Interval for the Variance (Mean Known) | p. 195 |

Confidence Interval for the Variance (Mean Unknown) | p. 199 |

Statistical Tests | p. 200 |

Test for the Population Mean (Variance Known) | p. 202 |

Test for the Population Mean (Variance unknown and N[greater than or equal]60) | p. 205 |

Test for the Population Mean (Variance Unknown and N[60) | p. 205 |

Test for the Population Variance (N[less than or equal]100) | p. 207 |

Test for the Population Variance (N]100) | p. 210 |

Problems | p. 212 |

Fitting Probability Models to Data | p. 215 |

Empirical Distributions | p. 215 |

The Frequency Histogram from Observed Data | p. 215 |

The Expected Frequency Histogram from a Theoretical Distribution | p. 218 |

The Empirical Cumulative Distribution Function | p. 222 |

Statistical Tests for Goodness of Fit | p. 226 |

The Chi Squared Goodness of Fit Test | p. 226 |

Problems | p. 234 |

Regression Analysis | p. 237 |

Statistical Measures Between Two Random Variables | p. 237 |

The Sample Covariance and Sample Correlation Coefficient | p. 237 |

The Least-Squares Straight Line | p. 239 |

Fitting a Straight Line through the Scatter Diagram | p. 239 |

Non-Linear Curves Reducible to Straight Lines | p. 243 |

Confidence Intervals of the Regression Model | p. 245 |

Confidence Interval of the Slope a ([sigma superscript 2 subscript X] Unknown, N[60) | p. 246 |

Confidence Interval of the Slope a ([sigma superscript 2 subscript X] Unknown, 60[less than or equal]N) | p. 247 |

Confidence Interval of the Predicted Y ([sigma superscript 2 subscript X] Unknown, N[60) | p. 247 |

Confidence Interval of the Predicted Y ([sigma superscript 2 subscript X] Unknown, 60[less than or equal]N) | p. 248 |

Confidence Interval of the Intercept b ([sigma superscript 2 subscript X] Unknown, 60[less than or equal]N) | p. 249 |

Confidence Interval of the Intercept b ([sigma superscript 2 subscript X] Unknown, 60[less than or equal]N) | p. 249 |

Problems | p. 255 |

Reliability of Engineering Systems | p. 257 |

The Concept of Reliability | p. 257 |

Time Reliability | p. 259 |

Reliability of Systems | p. 267 |

Systems in Series | p. 267 |

Systems in Parallel | p. 270 |

Hybrid Systems | p. 272 |

Engineering Models of Failure | p. 272 |

The Weibull Failure Model | p. 272 |

Fitting Data to a Weibull Failure Model | p. 275 |

Problems | p. 278 |

Design of Engineering Experiments | p. 281 |

The Concept of Statistical Experiment Design | p. 281 |

Estimating the Population Mean from Limited Sampling | p. 283 |

Estimating the Required Number of Measurements | p. 286 |

Pre-Specified Variance | p. 287 |

Pre-Specified Margin of Error | p. 288 |

Problems | p. 292 |

Experiments and Tests for Two or More Populations | p. 293 |

Comparison of Parameters of Two Populations | p. 293 |

Test for the Comparison of Means of Two Populations (Variances [sigma superscript 2 subscript X subscript 1] and [sigma superscript 2 subscript X subscript 2] are Known) | p. 293 |

Test for the Comparison of Means of Two Populations ([sigma superscript 2 subscript X subscript 1] and [sigma superscript 2 subscript X subscript 2] are Unknown, [sigma superscript 2 subscript X subscript 1] =[sigma superscript 2 subscript X subscript 2], 60[less than or equal]N[subscript 1] and 60[less than or equal]N[subscript 2]) | p. 295 |

Test for the Comparison of Means of Two Populations ([sigma superscript 2 subscript X subscript 1] and [sigma superscript 2 subscript X subscript 2] are Unknown, [sigma superscript 2 subscript X subscript 1] [not equal] [sigma superscript 2 subscript X subscript 2], 60[less than or equal]N[subscript 1] and and 60[less than or equal]N[subscript 2]) | p. 296 |

Test for the Comparison of Means of Two Populations ([sigma superscript 2 subscript X subscript 1] and [sigma superscript 2 subscript X subscript 2] are Unknown, [sigma superscript 2 subscript X subscript 1] = [sigma superscript 2 subscript X subscript 2], and N[subscript 1][60 or N[subscript 2][60) | p. 296 |

Test for the Comparison of Means of Two Populations ([sigma superscript 2 subscript X subscript 1] and [sigma superscript 2 subscript X subscript 2] are Unknown, [sigma superscript 2 subscript X subscript 1] [not equal] [sigma superscript 2 subscript X subscript 2], and N[subscript 1][60 or N[subscript 2][60) | p. 297 |

Test for the Comparison of Variances of Two Normal Populations | p. 299 |

Comparison of Means of Two or More Populations: Single Factor Analysis of Variance (Anova) | p. 302 |

Box Plots and the Logic Behind Anova | p. 302 |

Test for the Comparison of Means of More than Two Normal Populations | p. 307 |

Problems | p. 314 |

Stochastic Processes | p. 317 |

The Concept of a Stochastic Process | p. 317 |

First and Second-Order Statistics | p. 322 |

Density Function and Cumulative Distribution Function | p. 322 |

Mean, and Correlation Functions | p. 324 |

Stationarity | p. 334 |

The Correlogram | p. 335 |

Transformations of the Correlation Function: The Spectral Density | p. 340 |

Some Theoretical Stochastic Processes | p. 347 |

The Random Walk Process | p. 347 |

The Brownian Motion Process | p. 348 |

The White Gaussian Noise Process | p. 351 |

Time Series Analysis | p. 353 |

Time Average Versus Ensemble Properties | p. 354 |

Deterministic Trend | p. 355 |

Periodicity | p. 356 |

Models for the Random Component | p. 357 |

Problems | p. 362 |

Stochastic Differential Equations | p. 365 |

The Origin of Stochastic Differential Equations | p. 365 |

Stochastic Continuity, Differentiation, and Integration | p. 368 |

Mean Square Continuity | p. 368 |

Stochastic Differentiation | p. 369 |

Stochastic Integrals | p. 373 |

Solving Applied Stochastic Differential Equations | p. 376 |

Differential Equations with Random Initial Conditions | p. 379 |

Differential Equations with Random Forcing Functions | p. 391 |

Solution of Random Equations with Decomposition | p. 398 |

Solving Non-Linear Differential Equations | p. 402 |

Differential Equations with Random Coefficients | p. 409 |

Problems | p. 412 |

Conclusion | p. 415 |

Tables | p. 417 |

Cumulative Areas under the Standard Normal Probability Density Function | p. 417 |

Abscissa Values Corresponding to Areas under the Student's t Density Function with m Degrees of Freedom | p. 418 |

Abscissa Values Corresponding to Areas under the Chi-Squared Density Function with m Degrees of Freedom | p. 420 |

Abscissa Values Corresponding to Areas under the F Density Function with and Degrees of Freedom | p. 422 |

Answers to Problems | p. 430 |

Bibliography | p. 444 |

Index | p. 450 |

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