Statistical Distributions (eBook)

Applications and Parameter Estimates

Nick T. Thomopoulos (Autor)

eBook Download: PDF

2017 | 1st ed. 2017
XVII, 172 Seiten
Springer International Publishing (Verlag)
978-3-319-65112-5 (ISBN)

Infos

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 provides a 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. 

Inhalt

Preface 6
Acknowledgments 7
Contents 8
About the Author 14
Chapter 1: Statistical Concepts 15
1.1 Introduction 15
1.1.1 Probability Distributions, Random Variables, Notation and Parameters 15
1.2 Fundamentals 17
1.3 Continuous Distribution 17
1.4 Discrete Distributions 19
1.5 Sample Data Basic Statistics 20
1.6 Parameter Estimating Methods 21
1.6.1 Maximum-Likelihood-Estimator (MLE) 22
1.6.2 Method-of-Moments (MoM) 22
1.7 Transforming Variables 22
1.7.1 Transform Data to Zero or Larger 22
1.7.2 Transform Data to Zero and One 23
1.7.3 Continuous Distributions and Cov 25
1.7.4 Discrete Distributions and Lexis Ratio 25
1.8 Summary 25
Chapter 2: Continuous Uniform 26
2.1 Fundamentals 26
2.2 Sample Data 28
2.3 Parameter Estimates from Sample Data 29
2.4 Parameter Estimates When No Data 30
2.5 When (a, b) Not Known 30
2.6 Summary 32
Chapter 3: Exponential 33
3.1 Fundamentals 33
3.2 Table Values 35
3.3 Memory-Less Property 36
3.4 Poisson Relation 37
3.5 Sample Data 38
3.6 Parameter Estimate from Sample Data 38
3.7 Parameter Estimate When No Data 39
3.8 Summary 41
Chapter 4: Erlang 42
4.1 Introduction 42
4.2 Fundamentals 42
4.3 Tables 43
4.4 Sample Data 46
4.5 Parameter Estimates When Sample Data 46
4.6 Parameter Estimates When No Data 47
4.7 Summary 48
Chapter 5: Gamma 50
5.1 Introduction 50
5.2 Fundamentals 50
5.3 Gamma Function 51
5.4 Cumulative Probability 51
5.5 Estimating the Cumulative Probability 53
5.6 Sample Data 55
5.7 Parameter Estimates When Sample Data 55
5.8 Parameter Estimate When No Data 56
5.9 Summary 58
Chapter 6: Beta 59
6.1 Introduction 59
6.2 Fundamentals 60
6.3 Standard Beta 60
6.4 Beta Has Many Shapes 61
6.5 Sample Data 63
6.6 Parameter Estimates When Sample Data 63
6.7 Regression Estimate of the Mean from the Mode 65
6.8 Parameter Estimates When No Data 66
6.9 Summary 67
Chapter 7: Weibull 69
7.1 Introduction 69
7.2 Fundamentals 69
7.3 Standard Weibull 70
7.4 Sample Data 72
7.5 Parameter Estimate of gamma When Sample Data 72
7.6 Parameter Estimate of (k1, k2) When Sample Data 73
7.7 Parameter Estimate When No Data 76
7.8 Summary 78
Chapter 8: Normal 79
8.1 Introduction 79
8.2 Fundamentals 79
8.3 Standard Normal 80
8.4 Hastings Approximations 81
8.5 Tables of the Standard Normal 82
8.6 Sample Data 84
8.7 Parameter Estimates When Sample Data 84
8.8 Parameter Estimates When No Data 85
8.9 Summary 86
Chapter 9: Lognormal 87
9.1 Introduction 87
9.2 Fundamentals 87
9.3 Lognormal Mode 88
9.4 Lognormal Median 88
9.5 Sample Data 89
9.6 Parameter Estimates When Sample Data 91
9.7 Parameter Estimates When No Data 92
9.8 Summary 94
Chapter 10: Left Truncated Normal 95
10.1 Introduction 95
10.2 Fundamentals 95
10.3 Standard Normal 96
10.4 Left-Truncated Normal 96
10.5 Cumulative Probability of t 97
10.6 Sample Data 101
10.7 Parameter Estimates When Sample Data 101
10.8 LTN in Inventory Control 103
10.9 Distribution Center in Auto Industry 104
10.10 Dealer, Retailer or Store 105
10.11 Summary 105
Chapter 11: Right Truncated Normal 106
11.1 Introduction 106
11.2 Fundamentals 106
11.3 Standard Normal 107
11.4 Right-Truncated Normal 107
11.5 Cumulative Probability of k 108
11.6 Mean and Standard Deviation of t 109
11.7 Spread Ratio of RTN 109
11.8 Table Values 109
11.9 Sample Data 112
11.10 Parameter Estimates When Sample Data 113
11.11 Estimate delta When RTN 113
11.12 Estimate the ?-Percent-Point of x 114
11.13 Summary 115
Chapter 12: Triangular 116
12.1 Introduction 116
12.2 Fundamentals 116
12.3 Standard Triangular 116
12.4 Triangular 117
12.5 Table Values on y 119
12.6 Deriving x? = ?-Percent-Point on x 120
12.7 Parameter Estimates When No Data 121
12.8 Summary 121
Chapter 13: Discrete Uniform 122
13.1 Introduction 122
13.2 Fundamentals 122
13.3 Lexis Ratio 123
13.4 Sample Data 124
13.5 Parameter Estimates When Sample Data 124
13.6 Parameter Estimates When No Data 125
13.7 Summary 126
Chapter 14: Binomial 127
14.1 Introduction 127
14.2 Fundamentals 127
14.3 Lexis Ratio 128
14.4 Normal Approximation 129
14.5 Poisson Approximation 130
14.6 Sample Data 133
14.7 Parameter Estimates with Sample Data 133
14.8 Parameter Estimates When No Data 133
14.9 Summary 134
Chapter 15: Geometric 135
15.1 Introduction 135
15.2 Fundamentals 135
15.3 Number of Failures 136
15.4 Sample Data 136
15.5 Parameter Estimate with Sample Data 136
15.6 Number of Trials 137
15.7 Sample Data 139
15.8 Parameter Estimate with Sample Data 139
15.9 Parameter Estimate When No Sample Data 139
15.10 Lexis Ratio 140
15.11 Memory Less Property 141
15.12 Summary 141
Chapter 16: Pascal 142
16.1 Introduction 142
16.2 Fundamentals 142
16.3 Number of Failures 143
16.4 Parameter Estimate When Sample Data 143
16.5 Parameter Estimate When No Data 144
16.6 Number of Trials 145
16.7 Lexis Ratio 146
16.8 Parameter Estimate When Sample Data 146
16.9 Summary 148
Chapter 17: Poisson 149
17.1 Introduction 149
17.2 Fundamentals 149
17.3 Lexis Ratio 150
17.4 Parameter Estimate When Sample Data 150
17.5 Parameter Estimate When No Data 150
17.6 Exponential Connection 151
17.7 Poisson with Multi Units 152
17.8 Summary 154
Chapter 18: Hyper Geometric 155
18.1 Introduction 155
18.2 Fundamentals 155
18.3 Parameter Estimate When Sample Data 156
18.4 Binomial Estimate 156
18.5 Summary 158
Chapter 19: Bivariate Normal 159
19.1 Introduction 159
19.2 Fundamentals 159
19.3 Bivariate Normal 160
19.4 Marginal Distributions 160
19.5 Conditional Distribution 161
19.6 Bivariate Standard Normal 161
19.7 Marginal Distribution 161
19.8 Conditional Distributions 162
19.9 Approximation to the Cumulative Joint Probability 162
19.10 Statistical Tables 169
19.11 Summary 169
Chapter 20: Bivariate Lognormal 170
20.1 Introduction 170
20.2 Fundamentals 170
20.3 Cumulative Probability 172
20.4 Summary 174
References 175

Produkt-Details

• Erscheint lt. Verlag: 10.10.2017
• Zusatzinfo: XVII, 172 p. 22 illus., 21 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Schlagworte: bivariate lognormal • bivariate normal • Distribution • Erlang • left-truncated normal • Maximum Likelihood Estimator • parameter estimates • Probability • right-truncated normal • spread ratio • statistical distributions • Weibull
• ISBN-10: 3-319-65112-9 / 3319651129
• ISBN-13: 978-3-319-65112-5 / 9783319651125