Statistical Distributions: Applications and Parameter Estimates

176 Pages·2017·3.47 MB·English

by  Nick T. Thomopoulos (auth.)

Nick T. Thomopoulos
Statistical
Distributions
Applications and Parameter Estimates
Statistical Distributions
Nick T. Thomopoulos
Statistical Distributions
Applications and Parameter Estimates
NickT.Thomopoulos
StuartSchoolofBusiness
IllinoisInstituteofTechnology
BurrRidge,IL,USA
ISBN978-3-319-65111-8 ISBN978-3-319-65112-5 (eBook)
DOI10.1007/978-3-319-65112-5
LibraryofCongressControlNumber:2017949365
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For my wife, my children, and my
grandchildren.
Preface
A statistical distribution is a mathematical function that defines the probable
occurrenceofarandomvariableoveritsadmissiblespace.Understandingstatistical
distributionsisafundamentalrequisitetoresearchersinalmostalldisciplines.The
informedresearcherwillselectthestatisticaldistributionthatbestfitsthedatainthe
studyathand.Thisbookgivesadescriptionofthegroupofstatisticaldistributions
that have ample application to studies in statistics and probability. 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
commonuse.Thisbookdescribeswhenandhowtoapplyeachofthedistributions
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 vari-
ables.Inmoststudies,theparametervaluesarenotknownapriori,andsampledata
is needed to estimate the 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 book is easy to read and includes many
examples to guide the reader; it will be a highly useful reference to anyone who
does statistical and probability analysis. This includes management scientists,
market researchers, engineers, mathematicians, physicists, chemists, economists,
socialscienceresearchers,andstudentsinmanydisciplines.
BurrRidge,IL,USA NickT.Thomopoulos
vii
Acknowledgments
Thanks especially to my wife, Elaine Thomopoulos, who encouraged me to write
thisbook,andwhogaveconsultationwheneverneeded.DanielSussmanassistedin
proofingthetext.Thanksalsotothemanypeoplewhohavehelpedandinspiredme
over the years, including some former Illinois Institute of Technology (Illinois
Tech) Ph.D. students. I can name only a few here: Emanuel Betinis (National
UniversityofHealthScience),FredBock(IITResearchInstitute),DickChiapetta
(Chiapetta and Welch), Al Endres (Tampa University), John Garofalakis (Patras
University), James Hall (Caywood Schiller Associates), Montira Jantaravareerat
(Illinois Tech), Arvid Johnson (St. Francis University), Carol Lindee (Panduit),
Anatol Longinow (Illinois Tech), Fotis Mouzakis (Frynon Research), George
Resnikoff(CaliforniaStateUniversity),andPaulSpirakis(PatrasUniversity).
ix
Contents
1 StatisticalConcepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 ProbabilityDistributions,RandomVariables,
NotationandParameters. . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ContinuousDistribution. .. . . . .. . . . .. . . . .. . . . .. . . . .. 3
1.4 DiscreteDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 SampleDataBasicStatistics. . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 ParameterEstimatingMethods.. . . . .. . . . . .. . . . .. . . . .. 7
1.6.1 Maximum-Likelihood-Estimator(MLE). . . . . . . . . . 8
1.6.2 Method-of-Moments(MoM). . . . . . . . . . . . . . . . . . . 8
1.7 TransformingVariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7.1 TransformDatatoZeroorLarger. . . . . . . . . . . . . . . 8
1.7.2 TransformDatatoZeroandOne. . . . . . . . . . . . . . . . 9
1.7.3 ContinuousDistributionsandCov. . . . . . . . . . . . . . . 11
1.7.4 DiscreteDistributionsandLexisRatio. . . . . . . . . . . 11
1.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 ContinuousUniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 ParameterEstimatesfromSampleData. . . . . . . . . . . . . . . . 16
2.4 ParameterEstimatesWhenNoData. . . . . . . . . . . . . . . . . . . 17
2.5 When(a,b)NotKnown. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Exponential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 TableValues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Memory-LessProperty. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
xi
xii Contents
3.4 PoissonRelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 ParameterEstimatefromSampleData. . . . . . . . . . . . . . . . . 26
3.7 ParameterEstimateWhenNoData. . . . . . . . . . . . . . . . . . . 27
3.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Erlang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . 35
4.6 ParameterEstimatesWhenNoData. . . . . . . . . . . . . . . . . . . 36
4.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Gamma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 GammaFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 CumulativeProbability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 EstimatingtheCumulativeProbability. . . . . . . . . . . . . . . . . 42
5.6 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.7 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . 44
5.8 ParameterEstimateWhenNoData. . . . . . . . . . . . . . . . . . . 45
5.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Beta. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . 49
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 StandardBeta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.4 BetaHasManyShapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.5 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.6 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . 53
6.7 RegressionEstimateoftheMeanfromtheMode. . . . . . . . . 55
6.8 ParameterEstimatesWhenNoData. . . . . . . . . . . . . . . . . . . 56
6.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Weibull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.3 StandardWeibull. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.4 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.5 ParameterEstimateofγWhenSampleData. . . . . . . . . . . . . 62
7.6 ParameterEstimateof(k ,k )WhenSampleData. . . . . . . . 63
1 2
7.7 ParameterEstimateWhenNoData. . . . . . . . . . . . . . . . . . . 66
7.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Contents xiii
8 Normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.3 StandardNormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.4 HastingsApproximations. . .. . . .. . . . .. . . . .. . . .. . . . .. 71
8.5 TablesoftheStandardNormal. . .. . . . . .. . . . . . .. . . . . .. 72
8.6 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
8.7 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . 74
8.8 ParameterEstimatesWhenNoData. . . . . . . . . . . . . . . . . . . 75
8.9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9 Lognormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
9.3 LognormalMode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.4 LognormalMedian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.5 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.6 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . 81
9.7 ParameterEstimatesWhenNoData. . . . . . . . . . . . . . . . . . . 82
9.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10 LeftTruncatedNormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.3 StandardNormal. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 86
10.4 Left-TruncatedNormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10.5 CumulativeProbabilityoft. . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.6 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.7 ParameterEstimatesWhenSampleData. . . . . . . . . . . . . . . . 91
10.8 LTNinInventoryControl. . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.9 DistributionCenterinAutoIndustry. . . . . . . . . . . . . . . . . . . 94
10.10 Dealer,RetailerorStore. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
10.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11 RightTruncatedNormal. .. . . . . . . .. . . . . . .. . . . . . .. . . . . . . .. 97
11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2 Fundamentals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.3 StandardNormal. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 98
11.4 Right-TruncatedNormal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.5 CumulativeProbabilityofk. . . . . . . .. . . . . . . . . . . . . . . .. . 99
11.6 MeanandStandardDeviationoft. . . . . . . . . . . . . . . . . . . . . 100
11.7 SpreadRatioofRTN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.8 TableValues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
11.9 SampleData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103