Probability Distributions Used in Reliability Engineering

The book provides details on 22 probability distributions. Each distribution section provides a graphical visualization and formulas for distribution parameters, along with distribution formulas. Common statistics such as moments and percentile formulas are followed by likelihood functions and in many cases the derivation of maximum likelihood estimates. Bayesian non-informative and conjugate priors are provided followed by a discussion on the distribution characteristics and applications in reliability engineering. Preface......Page 6 Contents......Page 8 1. Fundamentals of Probability Distributions......Page 12 1.1.2. Interpretations of Probability Theory......Page 13 1.1.3. Laws of Probability......Page 14 1.1.5. Bayes’ Law......Page 15 1.1.6. Likelihood Functions......Page 16 1.1.7. Fisher Information Matrix......Page 17 1.2.3. Probability Density Function......Page 20 1.2.4. Cumulative Distribution Function......Page 22 1.2.5. Reliability Function......Page 23 1.2.9. Hazard Rate......Page 24 1.2.10. Cumulative Hazard Rate......Page 25 1.2.11. Characteristic Function......Page 26 1.2.12. Joint Distributions......Page 27 1.2.15. Bathtub Distributions......Page 28 1.2.16. Truncated Distributions......Page 29 1.2.17. Summary......Page 30 1.3.2. Moments of Distribution......Page 31 1.3.3. Covariance......Page 32 1.4.1. Probability Plotting Paper......Page 33 1.4.2. Total Time on Test Plots......Page 34 1.4.3. Least Mean Square Regression......Page 35 1.4.4. Method of Moments......Page 36 1.4.5. Maximum Likelihood Estimates......Page 37 1.4.6. Bayesian Estimation......Page 38 1.4.7. Confidence Intervals......Page 41 Related Distributions......Page 44 1.6.4. Complete Gamma Function ()......Page 45 1.6.6. Lower Incomplete Gamma Function (,)......Page 46 1.6.8. Trigamma Function ′......Page 47 1.7.4. Chi-Square Distribution ()......Page 48 1.7.6. Wishart Distribution (;,)......Page 49 1.8. Nomenclature and Notation......Page 50 2. Common Life Distributions......Page 51 2.1. Exponential Continuous Distribution......Page 52 2.2. Lognormal Continuous Distribution......Page 60 2.3. Weibull Continuous Distribution......Page 70 3. Bathtub Life Distributions......Page 79 3.1. 2-Fold Mixed Weibull Distribution......Page 80 3.2. Exponentiated Weibull Distribution......Page 87 3.3. Modified Weibull Distribution......Page 92 4. Univariate Continuous Distributions......Page 96 4.1. Beta Continuous Distribution......Page 97 4.2. Birnbaum Saunders Continuous Distribution......Page 104 4.3. Gamma Continuous Distribution......Page 110 4.4. Logistic Continuous Distribution......Page 119 4.5. Normal (Gaussian) Continuous Distribution......Page 126 4.6. Pareto Continuous Distribution......Page 136 4.7. Triangle Continuous Distribution......Page 142 4.8. Truncated Normal Continuous Distribution......Page 146 4.9. Uniform Continuous Distribution......Page 156 5. Univariate Discrete Distributions......Page 162 5.1. Bernoulli Discrete Distribution......Page 163 5.2. Binomial Discrete Distribution......Page 168 5.3. Poisson Discrete Distribution......Page 176 6. Bivariate and Multivariate Distributions......Page 183 6.1. Bivariate Normal Continuous Distribution......Page 184 6.2. Dirichlet Continuous Distribution......Page 192 6.3. Multivariate Normal Continuous Distribution......Page 198 6.4. Multinomial Discrete Distribution......Page 204 7. References......Page 212