Advances in combinatorial methods and applications to probability and statistics
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Birkhäuser
1997
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| Schriftenreihe: | Statistics for industry and technology
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| 245 | 1 | 0 | |a Advances in combinatorial methods and applications to probability and statistics |c N. Balakrishnan ed. |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1997 | |
| 300 | |a XXXIV, 562 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Statistics for industry and technology | |
| 500 | |a Literaturangaben | ||
| 600 | 1 | 7 | |a Mohanty, Gopal |d 1933- |0 (DE-588)119469243 |2 gnd |9 rswk-swf |
| 650 | 7 | |a Probabilités combinatoires |2 ram | |
| 650 | 7 | |a Statistique mathématique |2 ram | |
| 650 | 4 | |a Combinatorial probabilities | |
| 650 | 4 | |a Mathematical statistics | |
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| 650 | 0 | 7 | |a Wahrscheinlichkeitstheorie |0 (DE-588)4079013-7 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text | Contents
Preface xvii
Sri Gopal Mohanty—Life and Works xix
List of Contributors xxvii
List of Tables xxxi
List of Figures xxxiii
Part I—Lattice Paths and Combinatorial Methods
1 Lattice Paths and Faber Polynomials
Ira M. Gessel and Sangwook Ree 3
1.1 Introduction, 3
1.2 Faber Polynomials, 6
1.3 Counting Paths, 7
1.4 A Positivity Result, 10
1.5 Examples, 11
References, 13
2 Lattice Path Enumeration and Umbral Calculus
Heinrich Niederhausen 15
2.1 Introduction, 15
2.1.1 Notation, 16
2.2 Initial Value Problems, 16
2.2.1 The role of ex, 18
2.2.2 Piecewise affine boundaries, 18
2.2.3 Applications: Bounded paths, 19
2.3 Systems of Operator Equations, 20
2.3.1 Applications: Lattice paths with
several step directions, 21
2.4 Symmetric Sheffer Sequences, 21
2.4.1 Applications: Weighted left turns, 22
2.4.2 Paths inside a band, 23
viii Contents
2.5 Geometric Sheffer Sequences, 24
2.5.1 Applications: Crossings, 25
References, 26
3 The Enumeration of Lattice Paths With Respect
to Their Number of Turns
C. Krattenthaler 29
3.1 Introduction, 29
3.2 Notation, 31
3.3 Motivating Examples, 31
3.4 Turn Enumeration of (Single) Lattice Paths, 36
3.5 Applications, 44
3.6 Nonintersecting Lattice Paths and Turns, 47
References, 55
4 Lattice Path Counting, Simple Random Walk
Statistics, and Randomizations: An Analytic
Approach
Wolfgang Panny and Walter Katzenbeisser 59
4.1 Introduction, 59
4.2 Lattice Paths, 60
4.3 Simple Random Walks, 64
4.4 Randomized Random Walks, 70
References, 74
5 Combinatorial Identities: A Generalization of
Dougall s Identity
Erik Sparre Andersen and Mogens Esrom, Larsen 77
5.1 Introduction, 77
5.2 The Generalized Pfaff Saalschiitz Formula, 80
5.3 A Modified Pfaff Saalschiitz Sum of Type
11(4, 4,1)JV, 82
5.4 A Well Balanced 7/(5, 5,1)N Identity, 83
5.5 A Generalization of Dougall s Well Balanced
11(7, 7,1)N Identity, 85
References, 87
6 A Comparison of Two Methods for Random
Labelling of Balls by Vectors of Integers
Doron Zeilberger 89
6.1 First Way, 89
6.2 Second Way, 89
6.3 Variance and Standard Deviation, 91
Contents jx
6.4 Analysis of the Second Way, 92
References, 93
Part II—Applications to Probability Problems
7 On the Ballot Theorems
Lajos Takdcs 97
7.1 Introduction, 97
7.2 The Classical Ballot Theorem, 97
7.3 The Original Proofs of Theorem 7.2.1, 100
7.4 Historical Background, 102
7.5 The General Ballot Theorem, 104
7.6 Some Combinatorial Identities, 107
7.7 Another Extension of The Classical Ballot
Theorem, 109
References, 111
8 Some Results for Two Dimensional Random Walk
Endre Csdki 115
8.1 Introduction, 115
8.2 Identities and Distributions, 118
8.3 Pairs of LRW Paths, 120
References, 123
9 Random Walks on SL(2, F2) and Jacobi Symbols of
Quadratic Residues
Toshihiro Watanabe 125
9.1 Introduction, 125
9.2 Preliminaries, 126
9.3 A Calculation of the Character x(aM,m) and
Its Relation, 129
References, 133
10 Rank Order Statistics Related to a Generalized
Random Walk
Jagdish Saran and Sarita Rani 135
10.1 Introduction, 135
10.2 Some Auxiliary Results, 136
10.3 The Technique, 138
10.4 Definitions of Rank Order Statistics, 139
10.5 Distributions of N+*(a) and R+*n(a), 140
10.6 Distributions of A+ n(a) and Rf +n(a), 144
10.7 Distributions of N* n(a) and R*^n{a), 148
References, 151
x Contents
11 On a Subset Sum Algorithm and Its Probabilistic
and Other Applications
V. G. Voinov and M. S. Nikulin 153
11.1 Introduction, 153
11.2 A Derivation of the Algorithm, 154
11.3 A Class of Discrete Probability Distributions, 159
11.4 A Remark on a Summation Procedure When
Constructing Partitions, 160
References, 162
12 I and J Polynomials in a Potpourri of Probability
Problems
Milton Sobel 165
12.1 Introduction, 165
12.2 Guide to the Problems of this Paper, 166
12.3 Triangular Network with Common Failure
Probability q for Each Unit, 171
12.4 Duality Levels in a Square with Diagonals
That Do Not Intersect: Problem 12.5, 177
References, 183
13 Stirling Numbers and Records
N. Balakrishnan and V. B. Nevzorov 189
13.1 Stirling Numbers, 189
13.2 Generalized Stirling Numbers, 190
13.3 Stirling Numbers and Records, 193
13.4 Generalized Stirling Numbers and Records
in the Fa scheme, 195
13.5 Record Values from Discrete Distributions
and Generalized Stirling Numbers, 197
References, 198
Part III—Applications to Urn Models
14 Advances in Urn Models During The
Past Two Decades
Samuel Kotz and N. Balakrishnan 203
14.1 Introduction, 203
14.2 Polya Eggenberger Urns and Their Generalizations
and Modifications, 206
14.3 Generalizations of the Classical Occupancy
Model, 216
14.4 Ehrenfest Urn Model, 219
Contents xj
14.5 Polya Urn Model with a Continuum of Colors, 225
14.6 Stopping Problems in Urns, 226
14.7 Limit Theorems for Urns with Random
Drawings, 227
14.8 Limit Theorems for Sequential Occupancy, 228
14.9 Limit Theorems for Infinite Urn Models, 230
14.10 Urn Models with Indistinguishable Balls
(Bose Einstein Statistics), 231
14.11 Ewens Sampling Formula and Coalescent
Urn Models, 233
14.12 Reinforcement Depletion (Compartmental)
Urn Models, 237
14.13 Urn Models for Interpretation of Mathematical
and Probabilistic Concepts and Engineering and
Statistical Applications, 243
References, 247
15 A Unified Derivation of Occupancy and
Sequential Occupancy Distributions
Ch. A. Charalambides 259
15.1 Introduction, 259
15.2 Occupancy Distributions, 260
15.3 Sequential Occupancy Distributions, 267
References, 273
16 Moments, Binomial Moments and Combinatorics
Janos Galambos 275
16.1 Basic Relations, 275
16.2 Linear Inequalities in Sk, pr and qr, 277
16.3 A Statistical Paradox and an Urn Model
with Applications, 280
16.4 Quadratic Inequalities, 281
References, 283
Part IV—Applications to Queueing Theory
17 Nonintersecting Paths and Applications to
Queueing Theory
Walter Bohm, 287
17.1 Introduction, 287
17.2 Dissimilar Bernoulli Processes, 288
17.3 The r Node Series Jackson Network, 291
17.4 The Dummy Path Lemma for Poisson Processes, 295
17.5 A Special Variant of D/M/l Queues, 297
xii Contents
References, 299
18 Transient Busy Period Analysis of Initially
Non Empty M/G/l Queues—Lattice Path Approach
Kanwar Sen and Manju Agarwal 301
18.1 Introduction, 301
18.2 Lattice Path Approach, 304
18.3 Discretized M/C2/l Model, 304
18.3.1 Transition probabilities, 304
18.3.2 Counting of lattice paths, 307
18.3.3 Busy period probability, 310
18.4 Continuous M/C2/l Model, 312
18.5 Particular Cases, 313
References, 313
19 Single Server Queueing System with Poisson
Input: A Review of Some Recent Developments
J. Medhi 317
19.1 Introduction, 317
19.2 Exceptional Service for the First Unit in
Each Busy Period, 319
19.3 M/G/l With Random Setup Time S, 320
19.4 M/G/l System Under iV Policy, 322
19.5 M/G/l Under TV Policy and With Setup Time, 323
19.6 Queues With Vacation: M/G/l Queueing System
With Vacation, 324
19.7 M/G/l Vm System, 325
19.8 M/G/l Vm With Exceptional First Vacation, 326
19.9 M/G/l Vs System, 327
19.10 M/G/l System With Vacation and Under iV Policy
(With Threshold N), 328
19.11 Mx/G/l System With Batch Arrival, 332
19.12 Mx/G/l Under iV Policy, 332
19.13 Mx/G/l Vm and Mx/G/l Vs, 334
19.14 Mx/G/l Vacation Queues Under iV Policy, 334
19.15 Concluding Remarks, 335
References, 336
20 Recent Advances in the Analysis of Polling Systems
Diwakar Gupta and Yavuz Giinalay 339
20.1 Introduction, 339
20.2 Notations and Preliminaries, 342
20.3 Main Results, 346
Contents xiii
20.4 Some Related Models, 350
20.4.1 Customer routing, 350
20.4.2 Stopping only at a preferred station, 351
20.4.3 Gated or mixed service policy, 351
20.4.4 State dependent setups, 352
20.4.5 Periodic monitoring during idle period, 354
20.5 Insights, 355
20.6 Future Directions, 357
References, 357
Part V—Applications to Waiting Time Problems
21 Waiting Times and Number of Appearances of Events
in a Sequence of Discrete Random Variables
Markos V. Koutras 363
21.1 Introduction, 363
21.2 Definitions and Notations, 365
21.3 General Results, 366
21.4 Waiting Times and Number of Occurrences of
Delayed Recurrent Events, 370
21.5 Distribution of the Number of Success Runs in
a Two State Markov Chain, 373
21.5.1 Non overlapping success runs, 374
21.5.2 Success runs of length at least k, 376
21.5.3 Overlapping success runs, 378
21.5.4 Number of non overlapping windows of length at
most k containing exactly 2 successes, 378
21.6 Conclusions, 380
References, 380
22 On Sooner and Later Problems Between Success
and Failure Runs
Sigeo Aki 385
22.1 Introduction, 385
22.2 Number of Ocurrences of the Sooner Event Until
the Later Waiting Time, 387
22.3 Joint Distribution of Numbers of Runs, 397
References, 399
xiv Contents
23 Distributions of Numbers of Success Runs Until
the First Consecutive k Successes in Higher Order
Markov Dependent Trials
Katuomi Hirano, Sigeo Aki and Masayuki Uchida 401
23.1 Introduction, 401
23.2 Numbers of Success Runs in Higher Order
Markov Chain, 403
23.3 Case / m, 408
References, 409
24 On Multivariate Distributions of Various Orders
Obtained by Waiting for the r th Success Run
of Length k in Trials With Multiple Outcomes
Dem.etrios L. Antzoulakos and Andreas N. Philippou 411
24.1 Introduction, 412
24.2 Independent Trials, 413
24.3 Generalized Sequence of Order k, 420
References, 423
25 A Multivariate Negative Binomial Distribution of
Order k Arising When Success Runs are Allowed
to Overlap
Gregory A. Tripsiannis and Andreas N. Philippou 427
25.1 Introduction, 427
25.2 Multivariate Negative Binomial Distribution of
Order k, Type III, 429
25.3 Characteristics and Distributional Properties
of MNBk,ni(r; qi,..., qm), 431
References, 436
Part VI—Applications to Distribution Theory
26 The Joint Energy Distributions of the Bose Einstein
and of the Fermi Dirac Particles
/. Vincze and R. Toros 441
26.1 Introduction, 441
26.2 Derivation of the Joint Distribution and of the
Joint Entropy, 442
26.2.1 On the method, 442
26.2.2 Joint distribution of the number of particles in
energy intervals, 444 ,
26.3 Determination of the Limit Distributions, 447
26.4 Discussion, 448
Contents xv
References, 449
27 On Modified q Bessel Functions and Some
Statistical Applications
A. W. Kem,p 45X
27.1 Introduction, 451
27.2 Notation, 453
27.3 The Distribution of the Difference of Two Euler
Random Variables, 456
27.4 The Distribution of the Difference of Two Heine
Random Variables, 458
27.5 Comments on the Distribution of the Difference of
Two Generalized Euler Random Variables, 460
References, 462
28 A ^ Logarithmic Distribution
C. David Kemp 465
28.1 Introduction, 465
28.2 A q Logarithmic Distribution, 467
28.3 A Group Size Model for the Distribution, 469
References, 470
29 Bernoulli Learning Models: Uppuluri Numbers
K. G. Janardan 471
29.1 Introduction, 471
29.2 The General Model, 472
29.2.1 Special cases of the general probabilistic model, 474
29.3 Waiting Time Learning Models, 476
29.3.1 Special cases of waiting time learning models, 478
References, 480
Part VII—Applications to Nonparametric Statistics
30 Linear Nonparametric Tests Against Restricted
Alternatives: The Simple Tree Order and
The Simple Order
S. Chakraborti and W. Schaafsm,a 483
30.1 Introduction, 484
30.2 Background, 485
30.3 Objectives, 486
30.4 Exploration and Reformulation, 487
30.5 Test for the Simple Tree Problem, 487
30.5.1 Some particular cases, 490
30.5.2 Derivation of the MSSMP test, 493
xvi Contents
30.6 Test for the Simple Order Problem, 496
30.6.1 Derivation of the (A)MSSMP test, 497
30.6.2 Power comparisons, 499
30.7 Extending the Class of SMP Tests, 501
Appendix, 503
References, 505
31 Nonparametric Estimation of the Ratio of
Variance Components
M. Mahibbur Rahman and Z. Govindarajulu 507
31.1 Introduction, 507
31.2 Proposed Estimation Procedure, 510
31.3 Monte Carlo Comparison, 512
31.4 Adjustment for Bias, 514
References, 515
32 Limit Theorems for M Processes Via Rank
Statistics Processes
M. Huskovd 521
32.1 Introduction, 521
32.2 Case 9X = ¦ ¦ ¦ = 0n, 522
32.3 Change Point Alternatives, 526
References, 533
Author Index 535
Subject Index 545
|
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4143413-4 Aufsatzsammlung gnd-content |
| genre_facet | Aufsatzsammlung |
| id | DE-604.BV011446926 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T14:33:14Z |
| institution | BVB |
| isbn | 376433908X 081763908X |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007699824 |
| oclc_num | 36343429 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-384 DE-703 DE-824 DE-11 |
| owner_facet | DE-91G DE-BY-TUM DE-384 DE-703 DE-824 DE-11 |
| physical | XXXIV, 562 S. Ill., graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Statistics for industry and technology |
| spellingShingle | Advances in combinatorial methods and applications to probability and statistics Mohanty, Gopal 1933- (DE-588)119469243 gnd Probabilités combinatoires ram Statistique mathématique ram Combinatorial probabilities Mathematical statistics Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Kombinatorik (DE-588)4031824-2 gnd Bibliografie (DE-588)4006432-3 gnd |
| subject_GND | (DE-588)119469243 (DE-588)4056995-0 (DE-588)4079013-7 (DE-588)4031824-2 (DE-588)4006432-3 (DE-588)4143413-4 |
| title | Advances in combinatorial methods and applications to probability and statistics |
| title_auth | Advances in combinatorial methods and applications to probability and statistics |
| title_exact_search | Advances in combinatorial methods and applications to probability and statistics |
| title_full | Advances in combinatorial methods and applications to probability and statistics N. Balakrishnan ed. |
| title_fullStr | Advances in combinatorial methods and applications to probability and statistics N. Balakrishnan ed. |
| title_full_unstemmed | Advances in combinatorial methods and applications to probability and statistics N. Balakrishnan ed. |
| title_short | Advances in combinatorial methods and applications to probability and statistics |
| title_sort | advances in combinatorial methods and applications to probability and statistics |
| topic | Mohanty, Gopal 1933- (DE-588)119469243 gnd Probabilités combinatoires ram Statistique mathématique ram Combinatorial probabilities Mathematical statistics Statistik (DE-588)4056995-0 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Kombinatorik (DE-588)4031824-2 gnd Bibliografie (DE-588)4006432-3 gnd |
| topic_facet | Mohanty, Gopal 1933- Probabilités combinatoires Statistique mathématique Combinatorial probabilities Mathematical statistics Statistik Wahrscheinlichkeitstheorie Kombinatorik Bibliografie Aufsatzsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007699824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT balakrishnannarayanaswamy advancesincombinatorialmethodsandapplicationstoprobabilityandstatistics |