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El. knyga: Handbook of Statistical Distributions with Applications 2nd edition [Taylor & Francis e-book]

(University of Louisiana at Lafayette, USA)
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Handbook of Statistical Distributions with Applications 2nd editionDidesnis paveikslėlis
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Easy-to-Use Reference and Software for Statistical Modeling and Testing





Handbook of Statistical Distributions with Applications, Second Edition provides quick access to common and specialized probability distributions for modeling practical problems and performing statistical calculations. Along with many new examples and results, this edition includes both the authors StatCalc software and R codes to accurately and easily carry out computations.





New to the Second Edition















Major changes in binomial, Poisson, normal, gamma, Weibull, exponential, logistic, Laplace, and Pareto distributions





Updated statistical tests and intervals based on recent publications in statistical journals





Enhanced PC calculator StatCalc with electronic help manuals





R functions for cases where StatCalc is not applicable, with the codes available online





This highly praised handbook integrates popular probability distribution models, formulas, applications, and software to help you compute a variety of statistical intervals. It covers probability and percentiles, algorithms for random number generation, hypothesis tests, confidence intervals, tolerance intervals, prediction intervals, sample size determination, and much more.
List of Figures
xviii
List of Tables
xix
Symbols xx
Preface to the First Edition xxiii
Preface to the Second Edition xxv
1 StatCale
1(8)
1.1 Introduction
1(4)
1.2 Contents of Stat Cale
5(4)
2 Preliminaries
9(24)
2.1 Random Variables and Expectations
9(2)
2.2 Moments and Other Functions
11(4)
2.2.1 Measures of Central Tendency
12(1)
2.2.2 Moments
12(1)
2.2.3 Measures of Variability
13(1)
2.2.4 Measures of Relative Standing
13(1)
2.2.5 Other Measures
14(1)
2.2.6 Moment Generating Function
14(1)
2.3 Some Functions Relevant to Reliability
15(1)
2.4 Model Fitting
16(1)
2.4.1 Q Q Plot
16(1)
2.4.2 The Chi-Square Goodness-of-Fit Test
17(1)
2.5 Methods of Estimation
17(1)
2.6 Inference
18(5)
2.6.1 Hypothesis Testing
19(3)
2.6.2 Interval Estimation
22(1)
2.7 Pivotal-Based Methods for Location-Scale Families
23(4)
2.7.1 Pivotal Quantities
24(1)
2.7.2 Generalized Pivotal Quantities (GPQ)
25(2)
2.8 Method of Variance Estimate Recovery
27(1)
2.9 Modified Normal-Based Approximation
28(1)
2.9.1 Linear Combination of Independent Random Variables
28(1)
2.9.2 Ratio of Two Independent Random Variables
29(1)
2.10 Random Number Generation
29(1)
2.11 Some Special Functions
30(3)
Discrete Distributions
33(104)
3 Discrete Uniform Distribution
35(2)
3.1 Description
35(1)
3.2 Moments
36(1)
4 Binomial Distribution
37(36)
4.1 Description
37(1)
4.2 Moments
38(1)
4.3 Probabilities, Percentiles, and Moments
38(3)
4.4 Proportion
41(8)
4.4.1 Tests
42(1)
4.4.2 Power and Sample Size Calculation
43(2)
4.4.3 Confidence Intervals
45(2)
4.4.4 Sample Size Calculation for a Given Precision
47(2)
4.5 Prediction Intervals
49(3)
4.6 Tolerance Intervals
52(2)
4.6.1 Equal-Tailed and Two-Sided Tolerance Intervals
52(1)
4.6.2 Tolerance Intervals Based on Approximate Quantiles
53(1)
4.7 Tests for the Difference between Two Proportions
54(4)
4.7.1 Approximate Tests
55(1)
4.7.2 Fisher's Exact Test
56(1)
4.7.3 Powers and Sample Size Calculation
56(2)
4.8 Two-Sample Confidence Intervals for Proportions
58(6)
4.8.1 Difference
58(2)
4.8.2 Relative Risk and Odds Ratio
60(4)
4.9 Confidence Intervals for a Linear Combination of Proportions
64(2)
4.10 Properties and Results
66(2)
4.10.1 Properties
66(1)
4.10.2 Relation to Other Distributions
67(1)
4.10.3 Approximations
68(1)
4.11 Random Number Generation
68(1)
4.12 Computation of Probabilities
69(4)
5 Hypergeometric Distribution
73(16)
5.1 Description
73(1)
5.2 Moments
74(1)
5.3 Probabilities, Percentiles, and Moments
74(4)
5.4 Point Estimation
78(1)
5.5 Test for the Proportion and Power Calculation
79(1)
5.6 Confidence Interval and Sample Size Calculation
80(2)
5.7 Test for Comparing Two Proportions
82(3)
5.8 Properties and Results
85(1)
5.9 Random Number Generation
85(1)
5.10 Computation of Probabilities
86(3)
6 Poisson Distribution
89(26)
6.1 Description
89(1)
6.2 Moments
90(2)
6.3 Probabilities, Percentiles, and Moments
92(1)
6.4 Model Fitting with Examples
92(2)
6.5 One-Sample Inference
94(1)
6.6 Test for the Mean
94(2)
6.7 Confidence Intervals for the Mean
96(2)
6.8 Prediction Intervals
98(1)
6.9 Tolerance Intervals
99(2)
6.10 Tests for Comparing Two Means and Power Calculation
101(3)
6.11 Confidence Intervals for the Ratio of Two Means
104(2)
6.12 Confidence Intervals for the Difference between Two Means
106(2)
6.13 Inference for a Weighted Sum of Poisson Means
108(1)
6.14 Properties and Results
109(1)
6.14.1 Properties
109(1)
6.14.2 Relation to Other Distributions
110(1)
6.14.3 Approximations
110(1)
6.15 Random Number Generation
110(1)
6.16 Computation of Probabilities
111(4)
7 Geometric Distribution
115(4)
7.1 Description
115(1)
7.2 Moments
116(1)
7.3 Probabilities, Percentiles, and Moments
116(1)
7.4 Properties and Results
117(1)
7.5 Random Number Generation
117(2)
8 Negative Binomial Distribution
119(10)
8.1 Description
119(1)
8.2 Moments
120(1)
8.3 Probabilities, Percentiles, and Moments
120(2)
8.4 Point Estimation
122(1)
8.5 Test for the Proportion
123(1)
8.6 Confidence Intervals for the Proportion
124(1)
8.7 Properties and Results
124(1)
8.7.1 Properties
124(1)
8.7.2 Relation to Other Distributions
125(1)
8.8 Random Number Generation
125(1)
8.9 Computation of Probabilities
126(3)
9 Logarithmic Series Distribution
129(8)
9.1 Description
129(1)
9.2 Moments
129(1)
9.3 Probabilities, Percentiles, and Moments
130(2)
9.4 Inferences
132(2)
9.4.1 Point Estimation
132(1)
9.4.2 Interval Estimation
133(1)
9.5 Properties and Results
134(1)
9.6 Random Number Generation
134(1)
9.7 Computation of Probabilities
135(2)
Continuous Distributions
137(246)
10 Continuous Uniform Distribution
139(4)
10.1 Description
139(1)
10.2 Moments
139(1)
10.3 Inferences
140(1)
10.4 Properties and Results
141(1)
10.5 Random Number Generation
141(2)
11 Normal Distribution
143(40)
11.1 Description
143(1)
11.2 Moments
144(1)
11.3 Probabilities, Percentiles, and Moments
145(4)
11.4 One-Sample Inference
149(10)
11.4.1 Test for the Mean and Power Computation
149(2)
11.4.2 Confidence Interval for the Mean
151(1)
11.4.3 Confidence Interval for the Coefficient of Variation and Survival Probability
152(3)
11.4.4 Prediction Intervals
155(2)
11.4.5 Test for Variance
157(2)
11.5 Two-Sample Inference
159(7)
11.5.1 Inference for the Ratio of Variances
159(1)
11.5.2 Inference for the Difference between Two Means
160(3)
11.5.3 Inference for the Difference between Two Means when Variances Are Unknown and Arbitrary
163(1)
11.5.4 Comparison of Two Coefficients of Variation
164(2)
11.6 Tolerance Intervals
166(4)
11.6.1 Two-Sided Tolerance Intervals
166(1)
11.6.2 One-Sided Tolerance Limits
167(2)
11.6.3 Equal-Tailed Tolerance Intervals
169(1)
11.6.4 Simultaneous Hypothesis Testing for Quantiles
169(1)
11.7 Inference Based on Censored Samples
170(7)
11.7.1 Confidence Intervals for the Mean and Variance
172(1)
11.7.2 Tolerance Intervals
173(2)
11.7.3 Inference Based on Type I Censored Samples
175(2)
11.8 Properties and Results
177(1)
11.9 Relation to Other Distributions
178(1)
11.10 Random Number Generation
178(2)
11.11 Computation of the Distribution Function
180(3)
12 Chi-Square Distribution
183(6)
12.1 Description
183(1)
12.2 Moments
183(1)
12.3 Probabilities, Percentiles, and Moments
184(1)
12.4 Applications
185(1)
12.5 Properties and Results
186(2)
12.5.1 Properties
186(1)
12.5.2 Relation to Other Distributions
187(1)
12.5.3 Approximations
187(1)
12.6 Random Number Generation
188(1)
12.7 Computation of the Distribution Function
188(1)
13 F Distribution
189(6)
13.1 Description
189(1)
13.2 Moments
189(2)
13.3 Probabilities, Percentiles, and Moments
191(1)
13.4 Properties and Results
191(2)
13.4.1 Identities
191(1)
13.4.2 Relation to Other Distributions
192(1)
13.4.3 Series Expansions
192(1)
13.4.4 Approximations
193(1)
13.5 Random Number Generation
193(1)
13.6 A Computational Method for Probabilities
194(1)
14 Student's t Distribution
195(8)
14.1 Description
195(1)
14.2 Moments
196(1)
14.3 Probabilities, Percentiles, and Moments
197(1)
14.4 Distribution of the Maximum of Several |t| Variables
197(2)
14.4.1 Applications
197(1)
14.4.2 Percentiles of Max {|t1|...,|tk|}
198(1)
14.5 Properties and Results
199(2)
14.5.1 Properties
199(1)
14.5.2 Relation to Other Distributions
199(1)
14.5.3 Series Expansions for Cumulative Probability
200(1)
14.6 Random Number Generation
201(1)
14.7 Computation of the Distribution Function
201(2)
15 Exponential Distribution
203(12)
15.1 Description
203(1)
15.2 Moments
203(1)
15.3 Probabilities, Percentiles, and Moments
204(1)
15.4 Estimation
205(1)
15.5 Confidence Intervals
205(1)
15.6 Prediction Intervals
206(1)
15.7 Tolerance Limits and Survival Probability
207(2)
15.8 Two-Sample Case
209(3)
15.8.1 Confidence Interval for Comparing Two Parameters
209(1)
15.8.2 Ratio of Scale Parameters
210(1)
15.8.3 Confidence Interval for the Difference between Two Means
210(2)
15.9 Properties and Results
212(1)
15.10 Random Number Generation
213(2)
16 Gamma Distribution
215(32)
16.1 Description
215(1)
16.2 Moments
216(1)
16.3 Probabilities, Percentiles, and Moments
216(2)
16.4 Applications
218(1)
16.5 Estimation of Parameters: Three-Parameter Case
219(1)
16.6 One-Sample Inference
219(10)
16.6.1 One-Sample Tests
220(3)
16.6.2 One-Sample Confidence Intervals
223(3)
16.6.3 Prediction Intervals, Tolerance Intervals, and Survival Probability
226(3)
16.7 Comparison of Several Gamma Distributions
229(6)
16.8 Properties and Results
235(1)
16.9 Random Number Generation
236(1)
16.10 Computational Method for Probabilities
237(1)
16.11 R Programs
238(9)
17 Beta Distribution
247(12)
17.1 Description
247(2)
17.2 Moments
249(1)
17.3 Probabilities, Percentiles, and Moments
249(1)
17.4 Inferences
250(1)
17.5 Applications with an Example
250(3)
17.6 Properties and Results
253(1)
17.6.1 An Identity and Recurrence Relations
253(1)
17.6.2 Relation to Other Distributions
253(1)
17.7 Random Number Generation
254(2)
17.8 Computation of the Distribution Function
256(3)
18 Noncentral Chi-Square Distribution
259(8)
18.1 Description
259(1)
18.2 Moments
259(1)
18.3 Probabilities, Percentiles, and Moments
260(1)
18.4 Applications
260(2)
18.5 Properties and Results
262(1)
18.5.1 Properties
262(1)
18.5.2 Approximations to Probabilities
262(1)
18.5.3 Approximations to Percentiles
263(1)
18.6 Random Number Generation
263(1)
18.7 Evaluating the Distribution Function
264(3)
19 Noncentral F Distribution
267(8)
19.1 Description
267(1)
19.2 Moments
267(2)
19.3 Probabilities, Percentiles, and Moments
269(1)
19.4 Applications
269(1)
19.5 Properties and Results
270(1)
19.5.1 Properties
270(1)
19.5.2 Approximations
270(1)
19.6 Random Number Generation
270(1)
19.7 Evaluating the Distribution Function
271(4)
20 Noncentral t Distribution
275(8)
20.1 Description
275(1)
20.2 Moments
275(1)
20.3 Probabilities, Percentiles, and Moments
276(1)
20.4 Applications
277(1)
20.5 Properties and Results
277(1)
20.5.1 Properties
277(1)
20.5.2 An Approximation
278(1)
20.6 Random Number Generation
278(1)
20.7 Evaluating the Distribution Function
279(4)
21 Laplace Distribution
283(12)
21.1 Description
283(1)
21.2 Moments
283(1)
21.3 Probabilities, Percentiles, and Moments
284(1)
21.4 Maximum Likelihood Estimators
285(2)
21.5 Confidence Intervals and Prediction Intervals
287(1)
21.6 Tolerance Intervals
288(2)
21.7 Applications with Examples
290(3)
21.8 Relation to Other Distributions
293(1)
21.9 Random Number Generation
293(2)
22 Logistic Distribution
295(10)
22.1 Description
295(1)
22.2 Moments
295(1)
22.3 Probabilities, Percentiles, and Moments
296(1)
22.4 Maximum Likelihood Estimators
296(3)
22.5 Confidence Intervals and Prediction Intervals
299(2)
22.6 Tolerance Intervals
301(1)
22.7 Applications
302(1)
22.8 Properties and Results
303(1)
22.9 Random Number Generation
303(2)
23 Lognormal Distribution
305(10)
23.1 Description
305(1)
23.2 Moments
305(1)
23.3 Probabilities, Percentiles, and Moments
306(1)
23.4 Maximum Likelihood Estimators
307(1)
23.5 Confidence Interval and Test for the Mean
307(2)
23.6 Methods for Comparing Two Means
309(2)
23.7 Tolerance Limits, Prediction Limits, and Survival Probability
311(2)
23.8 Applications
313(1)
23.9 Properties and Results
313(1)
23.10 Random Number Generation
314(1)
23.11 Calculation of Probabilities and Percentiles
314(1)
24 Pareto Distribution
315(6)
24.1 Description
315(1)
24.2 Moments
315(1)
24.3 Probabilities, Percentiles, and Moments
316(1)
24.4 Confidence Intervals
317(1)
24.5 Prediction Intervals and Tolerance Limits
318(1)
24.6 Applications
318(1)
24.7 Random Number Generation
319(2)
25 Weibull Distribution
321(10)
25.1 Description
321(1)
25.2 Moments
321(1)
25.3 Probabilities, Percentiles, and Moments
322(1)
25.4 Maximum Likelihood Estimators and Pivotal Quantities
323(1)
25.5 Confidence Intervals and Prediction Intervals
324(2)
25.6 One-Sided Tolerance Limits
326(1)
25.7 Survival Probability
326(1)
25.8 Prediction Limits
327(2)
25.9 Properties and Results
329(1)
25.10 Random Number Generation
330(1)
26 Extreme Value Distribution
331(12)
26.1 Description
331(1)
26.2 Moments
332(1)
26.3 Probabilities, Percentiles, and Moments
333(1)
26.4 Maximum Likelihood Estimators
333(1)
26.5 Confidence Intervals
334(3)
26.6 Prediction Interval, Tolerance Limits, and Survival Probability
337(1)
26.7 Applications
338(1)
26.8 Two-Sample Inference
339(2)
26.9 Properties and Results
341(1)
26.10 Random Number Generation
341(2)
27 Cauchy Distribution
343(4)
27.1 Description
343(1)
27.2 Moments
343(1)
27.3 Probabilities and Percentiles
344(1)
27.4 Inference
344(1)
27.4.1 Estimation Based on Sample Quantiles
345(1)
27.4.2 Maximum Likelihood Estimators
345(1)
27.5 Applications
345(1)
27.6 Properties and Results
346(1)
28 Inverse Gaussian Distribution
347(6)
28.1 Description
347(1)
28.2 Moments
347(1)
28.3 Probabilities, Percentiles, and Moments
348(1)
28.4 One-Sample Inference
349(1)
28.4.1 A Test for the Mean
349(1)
28.4.2 Confidence Interval for the Mean
350(1)
28.5 Two-Sample Inference
350(2)
28.5.1 Inferences for the Difference between Two Means
350(1)
28.5.2 Inferences for the Ratio of Two Means
351(1)
28.6 Random Number Generation
352(1)
29 Ftayleigh Distribution
353(4)
29.1 Description
353(1)
29.2 Moments
353(1)
29.3 Probabilities, Percentiles, and Moments
354(1)
29.4 Confidence Interval
354(1)
29.5 Prediction Intervals and One-Sided Tolerance Limits
355(1)
29.6 Relation to Other Distributions
356(1)
29.7 Random Number Generation
356(1)
30 Bivariate Normal Distribution
357(12)
30.1 Description
357(1)
30.2 Computing Probabilities
358(1)
30.3 Inferences on Correlation Coefficients
359(5)
30.4 Comparison of Two Correlation Coefficients
364(1)
30.5 Test and Confidence Interval for Variances
365(1)
30.6 Some Properties
366(1)
30.7 Random Number Generation
367(1)
30.8 A Computational Algorithm for Probabilities
368(1)
31 Some Nonparametric Methods
369(14)
31.1 Distribution of Runs
369(3)
31.2 Sign Test and Confidence Interval for the Median
372(2)
31.3 Wilcoxon Signed-Rank Test and Mann Whitney U Statistic
374(2)
31.4 Wilcoxon Rank-Sum Test
376(3)
31.5 Quantile Estimation and Nonparametric Tolerance Interval
379(4)
References 383(12)
Index 395
Kalimuthu Krishnamoorthy, Ph.D., is a professor of statistics and SLEMCO Professor of Science at the University of Louisiana at Lafayette. He is an elected fellow of the American Statistical Association and an associate editor of Communications in Statistics. He has published more than 100 articles relating to small sample inference, multivariate analysis, fiducial inference, and statistical methods for exposure data analysis.
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