Description
This book gives a description of the group of statistical distributions that have ample application to studies in statistics and probability.Understanding statistical distributions is fundamental for researchers in almost all disciplines. The informed researcher will select the statistical distribution that best fits the data in the study at hand. Some of the distributions are well known to the general researcher and are in use in a wide variety of ways. Other useful distributions are less understood and are not in common use. The book describes when and how to apply each of the distributions in research studies, with a goal to identify the distribution that best applies to the study. The distributions are for continuous, discrete, and bivariate random variables. In most studies, the parameter values are not known a priori, and sample data is needed to estimate parameter values. In other scenarios, no sample data is available, and the researcher seeks some insight that allows the estimate of the parameter values to be gained. This handbook of statistical distributions providesa working knowledge of applying common and uncommon statistical distributions in research studies. These nineteen distributions are: continuous uniform, exponential, Erlang, gamma, beta, Weibull, normal, lognormal, left-truncated normal, right-truncated normal, triangular, discrete uniform, binomial, geometric, Pascal, Poisson, hyper-geometric, bivariate normal, and bivariate lognormal. Some are from continuous data and others are from discrete and bivariate data. This group of statistical distributions has ample application to studies in statistics and probability and practical use in real situations. Additionally, this book explains computing the cumulative probability of each distribution and estimating the parameter values either with sample data or without sample data. Examples are provided throughout to guide the reader. Accuracy in choosing and applying statistical distributions is particularly imperative for anyone who does statistical and probability analysis, including management scientists, market researchers, engineers, mathematicians, physicists, chemists, economists, social science researchers, and students in many disciplines. Nick T. Thomopoulos, Ph.D., has degrees in business (B.S.) and in mathematics (M.A.) from the University of Illinois, and in industrial engineering (Ph.D.) from Illinois Institute of Technology (Illinois Tech). He was supervisor of operations research at International Harvester; senior scientist at Illinois Tech Research Institute; Professor in Industrial Engineering, and in the Stuart School of Business at Illinois Tech. He is the author of eleven books including Fundamentals of Queuing Systems (Springer), Essentials of Monte Carlo Simulation (Springer), Applied Forecasting Methods (Prentice Hall), and Fundamentals of Production, Inventory and the Supply Chain (Atlantic). He has published many papers and has consulted in a wide variety of industries in the United States, Europe and Asia. Dr. Thomopoulos has received honors over the years, such as the Rist Prize from the Military Operations Research Society for new developments in queuing theory; the Distinguished Professor Award in Bangkok, Thailand from the Illinois Tech Asian Alumni Association; and the Professional Achievement Award from the Illinois Tech Alumni Association. 1. Statistical Concepts 1.1 Introduction Probability Distributions, Random Variables, Notation and Parameters 1.2 Fundamentals 1.3 Continuous Distribution Admissible Range Probability Density Cumulative Distribution Complementary Probability Expected Value Variance and Standard Deviation Median Coefficient-of-Variation 1.4 Discrete Distributions Admissible Range Probability Function Cumulative Probability Complementary Probability Expected Value and Mean Variance and Standard Deviation Median Mode Lexis Ratio 1.5 Sample Data Basic Statistics 1.6 Parameter Estimating Methods Maximum-Likelihood-Estimator (MLE) Method-of-Moments (MoM) 1.7 Transforming Variables Transform Data to Zero or Larger Transform Data to Zero and One Continuous Distributions and Cov Discrete Distributions and Lexis Ratio 1.8 Summary 2. Continuous Uniform Fundamentals Sample Data Parameter Estimates from Sample Data Parameter Estimates when No Data When (a, b) Not Known Summary 3. Exponential Fundamentals Table Values Memory-Less Property Poisson Relation Sample Data Parameter Estimate from Sample Data Parameter Estimate when No Data Summary 4. Erlang Introduction Fundamentals Tables Sample Data Parameter Estimates when Sample Data Parameter Estimates when No Data Summary 5. Gamma Introduction Fundamentals Gamma Function Cumulative Probability Estimating the Cumulative Probability Sample Data Parameter Estimates when Sample Data Parameter Estimate when No Data Summary 6. Beta Introduction Fundamentals Standard Beta Beta has Many Shapes Sample Data Parameter Estimates when Sample Data Regression Estimate of the Mean from the Mode Parameter Estimates when No Data Summary 7. Weibull Introduction Fundamentals Standard Weibull Sample Data Parameter Estimate of when Sample Data Parameter Estimate of (k1, k2) when Sample Data Solving for k1 Solving for k2 Parameter Estimate when No Data Summary 8. Normal Introduction Fundamentals Standard Normal Hastings Approximations Approximation of F(z) from z Approximation of z from F(z) Tables of the Standard Normal Sample Data Parameter Estimates when Sample Data Parameter Estimates when No Data Summary 9. Lognormal Introduction Fundamentals Lognormal Mode Lognormal Median Sample Data Parameter Estimates when Sample Data Parameter Estimates when No Data Summary 10. Left Truncated Normal Introduction Fundamentals Standard Normal Sample Data Parameter Estimates when Sample Data LTN in Inventory Control Distribution Center in Auto Industry Dealer, Retailer or Store Summary 11. Right Truncated Normal Introduction Fundamentals Standard Normal Right-Truncated Normal Cumulative Probability of k Mean and Standard Deviation of t Spread Ratio of RTN Table Values Sample Data Parameter Estimates when Sample Data Estimate when RTN Estimate the -percent-point of x Summary 12. Triangular Introduction Fundamentals Standard Triangular Triangular Parameter Estimates when No Data Summary 13. Discrete Uniform Introduction Fundamentals Lexis Ratio Sample Data Parameter Estimates when Sample Data Parameter Estimates when No Data Summary 14. Binomial Introduction Fundamentals Lexis Ratio Normal Approximation Poisson Approximation Sample Data Parameter Estimates with Sample Data Parameter Estimates when No Data Summary 15. Geometric Introduction Fundamentals Number of Failures Sample Data Parameter Estimate with Sample Data Number of Trials Sample Data Parameter Estimate with Sample Data Parameter Estimate when No Sample Data Lexis Ratio Memory Less Property Summary 16. Pascal Introduction Fundamentals Number of Failures Parameter Estimate when No Data Number of Trials Lexis Ratio Parameter Estimate when Sample Data Summary 17. Poisson Introduction Fundamentals Lexis Ratio Parameter Estimate when Sample Data Parameter Estimate when No Data Exponential Connection Summary 18. Hyper Geometric Introduction Fundamentals Parameter Estimate when Sample Data Binomial Estimate Summary 19. Bivariate Normal Introduction Fundamentals Bivariate Normal Marginal Distributions Conditional Distribution Bivariate Standard Normal Distributions Approximation to the Cumulative Joint Probability Statistical Tables Summary 20. Bivariate Lognormal Introduction Fundamentals Summary
