Fundamentals of probability: with stochastic processes
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2016]
|
| Edition: | Third edition |
| Series: | A Chapman & Hall book
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xvii, 614 Seiten Illustrationen |
| ISBN: | 9781498755016 |
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| 245 | 1 | 0 | |a Fundamentals of probability |b with stochastic processes |c Saeed Gharamani, Western New England University, Springfield, Massachusetts, USA |
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Record in the Search Index
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INTENT
PREFACE
XI
1 AXIOMS OF PROBABILITY
1.1 INTRODUCTION 1
1.2 SAMPLE SPACE AND EVENTS 3
1.3 AXIOMS OF PROBABILITY 10
1.4 BASIC THEOREMS 17
1.5 CONTINUITY OF PROBABILITY FUNCTION 25
1.6 PROBABILITIES 0 AND 1 27
1.7 RANDOM SELECTION OF POINTS FROM INTERVALS
REVIEW PROBLEMS 33
28
2 COMBINATORIAL METHODS
36
2.1
2.2
2.3
2.4
2.5
INTRODUCTION 36
COUNTING PRINCIPLE
36
NUMBER OF SUBSETS OF A SET
TREE DIAGRAMS
PERMUTATIONS 44
COMBINATIONS 50
STIRLING S FORMULA
REVIEW PROBLEMS
40
66
68
40
3 CONDITIONAL PROBABILITY AND INDEPENDENCE
71
3.1 CONDITIONAL PROBABILITY 71
REDUCTION OF SAMPLE SPACE 75
3.2 LAW OF MULTIPLICATION 80
3.3 LAW OF TOTAL PROBABILITY 83
3.4 BAYES FORMULA 94
3.5 INDEPENDENCE 102
3.6 APPLICATIONS OF PROBABILITY TO GENETICS 119
HARDY-WEINBERG LAW 123
SEX-LINKED GENES 125
REVIEW PROBLEMS 128
VI CONTENTS
^
4
DISTRIBUTION FUNCTIONS AND
131
DISCRETE RANDOM VARIABLES
4.1 RANDOM VARIABLES 131
4.2 DISTRIBUTION FUNCTIONS 135
4.3 DISCRETE RANDOM VARIABLES 144
4.4 EXPECTATIONS OF DISCRETE RANDOM VARIABLES 150
4.5 VARIANCES AND MOMENTS OF DISCRETE RANDOM VARIABLES 165
MOMENTS 170
4.6 STANDARDIZED RANDOM VARIABLES 173
REVIEW PROBLEMS 174
* 5 SPECIAL DISCRETE DISTRIBUTIONS 177
5.1 BERNOULLI AND BINOMIAL RANDOM VARIABLES 177
EXPECTATIONS AND VARIANCES OF BINOMIAL RANDOM VARIABLES 183
5.2 POISSON RANDOM VARIABLE 189
POISSON AS AN APPROXIMATION TO BINOMIAL 189
POISSON PROCESS 194
5.3 OTHER DISCRETE RANDOM VARIABLES 202
GEOMETRIC RANDOM VARIABLE 202
NEGATIVE BINOMIAL RANDOM VARIABLE 205
HYPERGEOMETRIC RANDOM VARIABLE 207
REVIEW PROBLEMS 214
* 6 CONTINUOUS RANDOM VARIABLES 218
6.1 PROBABILITY DENSITY FUNCTIONS 218
6.2 DENSITY FUNCTION OF A FUNCTION OF A RANDOM VARIABLE 227
6.3 EXPECTATIONS AND VARIANCES 232
EXPECTATIONS OF CONTINUOUS RANDOM VARIABLES 232
VARIANCES OF CONTINUOUS RANDOM VARIABLES 238
REVIEW PROBLEMS 244
* 7 SPECIAL CONTINUOUS DISTRIBUTIONS 246
7.1 UNIFORM RANDOM VARIABLE 246
7.2 NORMAL RANDOM VARIABLE 252
CORRECTION FOR CONTINUITY 255
7.3 EXPONENTIAL RANDOM VARIABLES 268
7.4 GAMMA DISTRIBUTION 274
CONTENTS
VN
75 BETA DISTRIBUTION 280
7.6 SURVIVAL ANALYSIS AND HAZARD FUNCTION
REVIEW PROBLEMS 290
286
* 8 BIVARIATE DISTRIBUTIONS
292
8.1 JOINT DISTRIBUTION OF TWO RANDOM VARIABLES 292
JOINT PROBABILITY MASS FUNCTIONS 292
JOINT PROBABILITY DENSITY FUNCTIONS 296
8.2 INDEPENDENT RANDOM VARIABLES 310
INDEPENDENCE OF DISCRETE RANDOM VARIABLES 310
INDEPENDENCE OF CONTINUOUS RANDOM VARIABLES 313
8.3 CONDITIONAL DISTRIBUTIONS 322
CONDITIONAL DISTRIBUTIONS: DISCRETE CASE 322
CONDITIONAL DISTRIBUTIONS: CONTINUOUS CASE 327
8.4 TRANSFORMATIONS OF TWO RANDOM VARIABLES 334
REVIEW PROBLEMS 342
* 9 MULTIVARIATE DISTRIBUTIONS
346
9.1 JOINT DISTRIBUTION OF N 2 RANDOM VARIABLES
JOINT PROBABILITY MASS FUNCTIONS 346
JOINT PROBABILITY DENSITY FUNCTIONS 354
RANDOM SAMPLE 358
9.2 ORDER STATISTICS 363
9.3 MULTINOMIAL DISTRIBUTIONS 369
REVIEW PROBLEMS 373
346
* 10 MORE EXPECTATIONS AND VARIANCES
375
10.1 EXPECTED VAL UES OF SUMS OF RANDOM VAN ABLES
PATTERN APPEARANCE 382
10.2 COVARIANCE 388
10.3 CORRELATION 402
10.4 CONDITIONING ON RANDOM VARIABLES 407
10.5 BIVARIATE NORMAL DISTRIBUTION 421
REVIEW PROBLEMS 425
375
VIII CONTENTS
^,
11
SUMS OF INDEPENDENT RANDOM 428
VARIABLES AND LIMIT THEOREMS
11.1 MOMENT-GENERATING FUNCTIONS 428
11.2 SUMS OF INDEPENDENT RANDOM VARIABLES 438
11.3 MARKOV AND CHEBYSHEV INEQUALITIES 446
CHEBYSHEV S INEQUALITY AND SAMPLE MEAN 450
11.4 LAWS OF LARGE NUMBERS 455
PROPORTION VERSUS DIFFERENCE IN COIN TOSSING 463
11.5 CENTRAL LIMIT THEOREM 466
REVIEW PROBLEMS 475
P- 12 STOCHASTIC PROCESSES 478
12.1 INTRODUCTION 478
12.2 MORE ON POISSON PROCESSES 479
WHAT IS A QUEUING SYSTEM? 490
PASTA: POISSON ARRIVALS SEE TIME AVERAGE 491
12.3 MARKOV CHAINS 494
CLASSIFICATIONS OF STATES OF MARKOV CHAINS 503
ABSORPTION PROBABILITY 513
PERIOD 517
STEADY-STATE PROBABILITIES 518
12.4 CONTINUOUS-TIME MARKOV CHAINS 530
STEADY-STATE PROBABILITIES 536
BIRTH AND DEATH PROCESSES 539
12.5 BROWNIAN MOTION 548
FIRST PASSAGE TIME DISTRIBUTION 555
THE MAXIMUM OF A BROWNIAN MOTION 556
THE ZEROS OF BROWNIAN MOTION 556
BROWNIAN MOTION WITH DRIFT 559
GEOMETRIC BROWNIAN MOTION 560
REVIEW PROBLEMS 563
* 13 SIMULATION 568
13.1 INTRODUCTION 568
13.2 SIMULATION OF COMBINATORIAL PROBLEMS 572
13.3 SIMULATIONOF CONDITIONAL PROBABILITIES 575
13.4 SIMULATIONOF RANDOM VARIABLES 578
13.5 MONTE CARLO METHOD 586
APPENDIX TABLES
CONTENTS IX
590
ANSWERS TO ODD-NUMBERED EXERCISES
INDEX
594
605
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| spelling | Ghahramani, Saeed Verfasser aut Fundamentals of probability with stochastic processes Saeed Gharamani, Western New England University, Springfield, Massachusetts, USA Third edition Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2016] © 2016 xvii, 614 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s 2\p DE-604 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029048103&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ghahramani, Saeed Fundamentals of probability with stochastic processes Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4064324-4 (DE-588)4123623-3 |
| title | Fundamentals of probability with stochastic processes |
| title_auth | Fundamentals of probability with stochastic processes |
| title_exact_search | Fundamentals of probability with stochastic processes |
| title_full | Fundamentals of probability with stochastic processes Saeed Gharamani, Western New England University, Springfield, Massachusetts, USA |
| title_fullStr | Fundamentals of probability with stochastic processes Saeed Gharamani, Western New England University, Springfield, Massachusetts, USA |
| title_full_unstemmed | Fundamentals of probability with stochastic processes Saeed Gharamani, Western New England University, Springfield, Massachusetts, USA |
| title_short | Fundamentals of probability |
| title_sort | fundamentals of probability with stochastic processes |
| title_sub | with stochastic processes |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Wahrscheinlichkeitsrechnung Lehrbuch |
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