Probability and stochastic modeling:
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Boca Raton, Fla. [u.a.]
CRC Press
2013
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| Series: | A Chapman & Hall book
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| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | XVII, 490 S. graph. Darst. |
| ISBN: | 9781439872062 1439872066 |
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| 245 | 1 | 0 | |a Probability and stochastic modeling |c Vladimir I. Rotar |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press |c 2013 | |
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| adam_text | Titel: Probability and stochastic modeling
Autor: Rotarʹ, Vladimir I
Jahr: 2013
Contents
Flag-signs in the margins designate a route: either Route 1 (which is entirely contained in
Route 2) or Route 2; see the Preface and Introduction for a description of these routes. A
new flag-sign is posted only at the moment of a route change.
Introduction 1 î£J
Chapter 1. Basic Notions 3
1 Sample Space and Events........................... 3
1.1 Sample space............................. 3
1.2 Events................................. 5
1.3 Class of events............................ 7 UJ
2 Probabilities ................................. 8 0
2.1 Probability measure ......................... 8
2.2 Probabilities in discrete spaces.................... 8
2.3 Examples of probability measures in non-discrete spaces...... 11
2.4 Probability space........................... 12 UJ
2.5 Some properties of probability measures .............. 13 1¿1
2.6 Continuity of probability measures ................. 14 uJ
3 Counting Techniques............................. 16 v3
3.1 The basic counting principle..................... 16
3.2 Permutations............................. 16
3.3 Partitions into groups......................... 18
3.4 Samples................................ 18
3.4.1 Unordered samples or combinations ........... 18
3.4.2 Ordered samples...................... 20
3.5 Sampling and occupancy problems ................. 22 IÜ
3.6 Matching............................... 24
4 Exercises ................................... 26 UÜ
Chapter 2. Independence and Conditional Probability 31
1 Independence ................................. 31
1.1 Definitions and examples....................... 31
1.2 Independent trials........................... 34
1.2.1 Binomial scheme ..................... 34
1.2.2 Binomial trees....................... 35
1.2.3 Multinomial scheme ................... 36 12J
2 Conditioning ................................. 37 £3
2.1 Conditional probability........................ 37
IX
x Contents
2.2 Multiplication rule and probabilities on trees ............ 38
2.3 The law of total probability...................... 39
2.4 Random walk and ruin probability.................. 41
2.5 Choosing the best object....................... 44
2.6 Bayes formula............................ 45
2.7 The two-armed bandit problem.................... 46
EU 3 The Borei-Cantelli Theorem......................... 47
!£j 4 Exercises ................................... 50
Chapter 3. Discrete Random Variables 55
1 Random Variables and Vectors........................ 55
1.1 Random variables and their distributions............... 55
1.2 Random vectors and joint distributions ............... 60
1.2.1 Joint and marginal distributions.............. 60
1.2.2 Independent random variables .............. 62
1.2.3 Convolution........................ 64
1.2.4 Conditional distributions ................. 65
2 Expected Value................................ 66
2.1 Definitions and properties ...................... 66
2.2 A useful formula for the expectation of a non-negative integer-
ID valued random variable ....................... 72
2.3 Infinite expected values. Do they occur in models of real phenom-
ena? .................................. 72
X¿j 2.4 Jensen s inequality ......................... 74
3 Variance and Other Moments. Inequalities for Deviations.......... 75
3.1 Definitions and examples....................... 75
3.2 Properties of variance ........................ 78
3.3 Inequalities for deviations...................... 81
4 Some Basic Distributions........................... 83
4.1 The binomial distribution....................... 83
4.2 The multinomial distribution..................... 83
4.3 The geometric distribution...................... 84
4.4 The negative binomial distribution.................. 85
4.5 The Poisson distribution and theorem................ 87
4.5.1 Poisson approximation .................. 87
4.5.2 Proof of the Poisson theorem............... 91
12J 4.5.3 Modeling a process of arrivals .............. 91
yS 4.5.4 Summation property ................... 92
4.5.5 Marked Poisson random variables ............ 93
5 Convergence of Random Variables. The Law of Large Numbers...... 95
5.1 Convergence of random variables.................. 95
5.2 The law of large numbers (LLN)................... 96
5.3 On a proof of the strong LLN .................... 98
UJ 5.4 More on convergence almost surely and in probability ....... 99
V¿i 6 Conditional Expectation ........................... 101
Contents xi
6.1 Conditional expectation given a random variable.......... 101
6.2 The formula for total expectation and other properties of condi-
tional expectation........................... 104
6.3 A formula for variance........................ 107 ©
6.4 Conditional expectation and least squares prediction........ 108
7 Exercises ...................................108 feJ
Chapter 4. Generating Functions. Branching Processes. 13
Random Walk Revisited 117
1 Branching Processes .............................117
2 Generating Functions.............................119
3 Branching Processes Revisited........................122
4 More on Random Walk............................123
4.1 Revisiting the initial state.......................123
4.2 Symmetric random walk: Ties and leadership............126
4.2.1 The number of ties.....................126
4.2.2 The number of leadership intervals. The arcsine law ... 127
5 Exercises ...................................130
Chapter 5. Markov Chains 131 S3
1 Definitions and Examples. Probability Distributions of Markov Chains . . 131
1.1 A preliminary example........................131
1.2 A general framework.........................132
1.3 Transition in many steps.......................136
2 First Step Analysis. Passage Times .....................139 ßj
2.1 An example..............................139
2.2 The case of absorbing states.....................140
2.3 How to find the mean passage time .................143
3 Variables Defined on a Markov Chain....................144
3.1 A preliminary example........................144
3.2 General model............................145
4 Ergodicity and Stationary Distributions ...................147 li!J
4.1 Ergodicity property..........................147
4.2 Stationary distribution........................150
4.3 Limiting distributions for chains with an infinite number of states . 152 0
4.4 Proofs of Theorems 4 and 7.....................153
4.4.1 Proof of Theorem 4....................153
4.4.2 Proof of Theorem 7....................155
4.4.3 An important remark ...................156
5 A Classification of States and Ergodicity...................156
5.1 Classes of states ...........................156
5.2 Recurrence property.........................157
5.3 Recurrence and travel times.....................160
5.4 Recurrence and ergodicity......................160
6 Exercises ...................................161 z
xii Contents
Chapter 6. Continuous Random Variables 167
1 Continuous Distributions........................... 167
1.1 Probability density function..................... 167
1.2 Cumulative distribution functions.................. 170
1.3 Expected value............................ 172
1.4 Transformations of random variables ................ 174
1.4.1 The distributions of transformations............ 174
1.4.2 Expectations........................ 174
1.5 Linear transformation of random variables. Normalization..... 177
2 Some Basic Distributions........................... 178
2.1 The uniform distribution....................... 178
2.2 The exponential distribution..................... 179
2.3 The r(gamma)-distribution...................... 181
2.4 The normal distribution........................ 183
3 Continuous Multivariate Distributions.................... 185
4 Sums of Independent Random Variables................... 188
4.1 Convolution in the continuous case ................. 188
4.2 Two more classical examples..................... 191
4.3 An additional remark regarding convolutions:
E3 Stable distributions.......................... 193
¿£Í 5 Conditional Distributions and Expectations ................. 194
5.1 Conditional density and expectation................. 194
5.2 Conditioning. Formulas for total probability and expectation .... 197
6 Exercises...................................202
Chapter 7. Distributions in the General Case. Simulation 209
1 Distribution Functions ............................209
1.1 Properties of distribution functions..................209
1.2 Three basic propositions.......................210
1.3 Decomposing into discrete and continuous components.......212
2 Expected Values................................214
2.1 Definitions..............................214
12J 2.2 A formula for the expectation of a positive random variable . . . .216
2.3 Proof of Proposition 6........................217
2.4 A generalization: Infinite expectations................218
2.5 On a general definition of expectation................218
3 On Convergence in Distribution and Probability...............219
3.1 Weak convergence..........................219
3.2 Comparison with convergence in probability ............220
4 Simulation ..................................221
4.1 Simulation of discrete random variables...............222
4.2 The cumulative distribution function method............223
4.3 Special methods for particular distributions.............223
4.4 On the Monte Carlo Method.....................225
5 Histograms..................................226
ra
Contents xiii
6 Exercises ...................................227
Chapter 8. Moment Generating Functions 229 UU
1 Definitions and Properties...........................229
1.1 Laplace transform ..........................229
1.2 The moment generating functions of some basic distributions . . .232
1.2.1 The binomial distribution.................232
1.2.2 The geometric and negative binomial distributions .... 232
1.2.3 The Poisson distribution..................232
1.2.4 The uniform distribution..................233
1.2.5 The exponential and gamma distributions.........233
1.2.6 The normal distribution..................233
1.3 Moment generating function and moments .............234
2 Some Examples of Applications.......................235
2.1 Sums of normal and gamma random variables............235
2.2 An example where the moment generating functions are rational
functions...............................236
2.3 The sum of a random number of random variables .........236
2.4 Moment generating functions in Finance ..............238
3 Exponential or Bernstein-Chernoff s Bounds................242
3.1 Main idea...............................242
3.2 A bound for large deviations for sums of random variables.....242
3.3 Proof of Theorem 3..........................246
4 Exercises ...................................247
Chapter 9. The Central Limit Theorem v3
for Independent Random Variables 249
1 The Central Limit Theorem (CLT) for Independent and Identically Dis-
tributed Random Variables ..........................249
1.1 A theorem and examples.......................249
1.2 On a proof of the CLT........................253 E3
1.3 Why the limiting distribution is normal: The stability property . . .254
2 The CLT for Independent Variables in the General Case...........256
2.1 The Lyapunov fraction and the Berry-Esseen theorem.......256
2.2 Necessary and sufficient conditions for normal convergence .... 260 __
3 Exercises ...................................262 UJ
Chapter 10. Covariance Analysis. UJ
The Multivariate Normal Distribution.
The Multivariate Central Limit Theorem 265
1 Covariance and Correlation..........................265
1.1 Covariance..............................265
1.2 The Cauchy-Schwarz inequality...................267
1.3 Correlation..............................268
1.4 A new perspective: Geometric interpretation ............269
ID
xiv Contents
1.5 The prediction problem: Linear regression..............271
2 Covariance Matrices and Some Applications.................273
2.1 Covariance matrices.........................273
2.2 Principal components method....................275
2.3 A model of portfolio optimization..................277
3 The Multivariate Normal Distribution ....................282
3.1 Definitions and properties......................282
3.2 The %2-distribution..........................285
3.3 The multivariate central limit theorem................288
4 Exercises ...................................291
Chapter 11. Maxima and Minima of Random Variables.
Elements of Reliability Theory.
Hazard Rate and Survival Probabilities 295
1 Maxima and Minima of Random Variables.
Reliability Characteristics...........................295
1.1 Main representations and examples .................295
1.2 More about reliability characteristics.................297
2 Limit Theorems for Maxima and Minima..................299
2.1 Preliminary limit theorems: Convergence to bounds.........299
2.2 A limit theorem for the minima of bounded variables........300
2.3 A limit theorem for the maxima of bounded variables........302
2.4 A limit theorem for unbounded variables ..............303
fcj 3 Hazard Rate. Survival Probabilities .....................305
3.1 A definition and properties......................305
3.2 Remaining lifetimes ........................309
4 Exercises...................................311
Chapter 12. Stochastic Processes: Preliminaries 315
1 General Definition ..............................315
2 Processes with Independent Increments ...................317
12-i 3 Brownian Motion...............................318
V¿j 4 Markov Processes ..............................321
UU 5 Representation and Simulation of Markov Processes in Discrete Time . . .322
i£j 6 Exercises...................................324
Chapter 13. Counting and Queuing Processes.
Birth and Death Processes: A General Scheme 325
1 Poisson Processes............................... 325
1.1 The homogeneous Poisson process.................. 325
uJ 1.2 The non-homogeneous Poisson process............... 328
1.3 Another perspective: Infinitesimal approach............. 330
v~ 2 Birth and Death Processes .......................... 333
2.1 A Markov queuing model with one server (M/M/l)......... 333
2.2 Proof of Proposition 2........................ 336
Contents xv
2.3 Birth and death processes: A general scheme............338 UJ
2.4 Limiting probabilities ........................339
2.5 Proof of Proposition 4........................343
Exercises ...................................344
.*
Chapter 14. Elements of Renewal Theory 349 UJ
1 Preliminaries .................................349
2 Limit Theorems................................350
3 Some Proofs..................................353
4 Exercises ...................................355
Chapter 15. Martingales in Discrete Time 357 IHJ
1 Definitions and Properties ..........................357
1.1 Two formulas of a general nature ..................357
1.2 Martingales: General properties and examples............358
1.3 Predictable sequences. Doob s decomposition............362
1.4 Martingale transform.........................364
2 Optional Time and Some Applications ...................365 UJ
2.1 Definitions and stopping property..................365
2.2 Wald s identity............................368
2.3 The ruin probability for the simple random walk..........370
3 Martingales and a Financial Market Model..................371
3.1 A pricing procedure .........................371
3.2 A static financial market model and the arbitrage theorem .....373
3.3 A dynamic model. Binomial trees..................377
4 Limit Theorems for Martingales.......................381
4.1 The strong LLN for martingales...................381
4.2 On convergence of random series ..................381
4.3 The central limit theorem.......................382
4.4 Kolmogorov s inequality.......................384
4.5 Proof of Theorem 9..........................385
4.6 Proof of Theorem 8..........................385
4.7 Proof of the LLN for independent and identically distributed ran-
dom variables ............................386
4.7.1 Sufficiency........................386
4.7.2 Necessity.........................387
5 Exercises...................................388 w¿J
Chapter 16. Brownian Motion and Martingales in Continuous Time 391 lîJ
1 Brownian Motion and Its Generalizations ..................391
1.1 Further properties of the standard Brownian motion.........391
1.1.1 Non-differentiability of trajectories............391
1.1.2 Brownian motion as an approximation:
The Donsker-Prokhorov invariance principle ......392
0
xvi Contents
1.1.3 The distribution of wt, hitting times, and the maximum
value of Brownian motion.................393
1.2 Brownian motion with drift and geometric Brownian motion .... 395
2 Martingales in Continuous Time.......................397
2.1 General properties and examples...................397
2.2 Risk neutral evaluation. Black-Scholes formula...........399
2.3 Optional stopping time and some applications............401
2.3.1 Definitions and examples.................401
2.3.2 The ruin probability for the Brownian motion with drift . 402
2.3.3 The distribution of the ruin time in the case of Brownian
motion...........................404
2.3.4 Hitting time for the Brownian motion with drift.....405
3 Exercises ...................................406
Chapter 17. More on Dependency Structures 409
1 Arrangement Structures and the Corresponding Dependencies .......410
1.1 Markov random fields........................411
1.2 Local dependency ..........................413
1.3 Point processes............................414
2 Measures of Dependency...........................416
2.1 Covariance functions.........................416
2.2 Mixing................................419
3 Limit Theorems for Dependent Random Variables..............420
3.1 A limit theorem for stationary sequences with strong mixing . . . .421
3.2 A limit theorem for the case of local dependency..........424
4 Symmetric Distributions. De Finetti s Theorem...............424
4.1 Conditional independency......................424
4.2 Symmetrically dependent or exchangeable random variables.
De Finetti s theorem.........................427
4.3 Limit theorems for exchangeable random variables.........430
4.3.1 The law of large numbers.................430
4.3.2 The central limit theorem.................431
4.3.3 Proofs...........................433
5 Exercises ...................................434
Chapter 18. Comparison of Random Variables.
Risk Evaluation 435
1 Some Particular Criteria ...........................435
1.1 Preference order...........................435
1.2 Several simple criteria........................436
1.2.1 Value-at-Risk (VaR)....................436
1.2.2 An important remark: Rather risk measures than criteria . 440
1.2.3 Tail conditional expectation or Tail-Value-at-Risk . . . .440
1.2.4 The mean-and-standard-deviation criterion........441
2 Expected Utility................................443
Contents xvii
2.1 Expected utility maximization (EUM)................443
2.2 Some particular utility functions. How to estimate utility functions 448
2.3 Risk aversion.............................450
3 Generalizations of the EUM Criterion....................453 0
3.1 Allais paradox............................453
3.2 Implicit or comparative utility....................454
3.3 Rank dependent expected utility...................456
4 Exercises ...................................457 !£J
Appendix. Tables. Some Facts from Calculus and the Theory of Interest 461
1 Tables.....................................461
2 Some Facts from Calculus ..........................464
2.1 The little o notation ........................464
2.2 Taylor s expansions..........................465
2.2.1 General expansion.....................465
2.2.2 Some particular expansions................466
2.3 Concavity...............................467
3 Some Facts from the Theory of Interest ...................468
3.1 Compounding interest........................468
3.2 Present value and discount......................469
References 471
Answers to Exercises 475
Index 485
|
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| owner_facet | DE-83 DE-824 DE-11 |
| physical | XVII, 490 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | CRC Press |
| record_format | marc |
| series2 | A Chapman & Hall book |
| spelling | Rotarʹ, Vladimir I. Verfasser (DE-588)134277945 aut Probability and stochastic modeling Vladimir I. Rotar Boca Raton, Fla. [u.a.] CRC Press 2013 XVII, 490 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Chapman & Hall book Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Probabilities. Stochastic models. (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Stochastisches Modell (DE-588)4057633-4 s DE-604 http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=4676412&custom_att_2=simple_viewer Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025767440&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rotarʹ, Vladimir I. Probability and stochastic modeling Stochastisches Modell (DE-588)4057633-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4057633-4 (DE-588)4079013-7 (DE-588)4123623-3 |
| title | Probability and stochastic modeling |
| title_auth | Probability and stochastic modeling |
| title_exact_search | Probability and stochastic modeling |
| title_full | Probability and stochastic modeling Vladimir I. Rotar |
| title_fullStr | Probability and stochastic modeling Vladimir I. Rotar |
| title_full_unstemmed | Probability and stochastic modeling Vladimir I. Rotar |
| title_short | Probability and stochastic modeling |
| title_sort | probability and stochastic modeling |
| topic | Stochastisches Modell (DE-588)4057633-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Stochastisches Modell Wahrscheinlichkeitstheorie Lehrbuch |
| url | http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=4676412&custom_att_2=simple_viewer http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025767440&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT rotarʹvladimiri probabilityandstochasticmodeling |
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