ISBN-10:
0387004254
ISBN-13:
9780387004259
Pub. Date:
03/12/2003
Publisher:
Springer-Verlag New York, LLC
Applied Probability (Springer Texts in Statistics Series) / Edition 1

Applied Probability (Springer Texts in Statistics Series) / Edition 1

by Kenneth Lange

Hardcover

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Product Details

ISBN-13: 9780387004259
Publisher: Springer-Verlag New York, LLC
Publication date: 03/12/2003
Series: Springer Texts in Statistics Series
Edition description: New Edition
Pages: 322
Product dimensions: 9.21(w) x 6.14(h) x 0.75(d)

Table of Contents

Prefacev
1Basic Notions of Probability Theory1
1.1Introduction1
1.2Probability and Expectation1
1.3Conditional Probability6
1.4Independence8
1.5Distributions, Densities, and Moments9
1.6Convolution12
1.7Random Vectors13
1.8Multivariate Normal Random Vectors16
1.9Convergence18
1.10Problems20
2Calculation of Expectations25
2.1Introduction25
2.2Indicator Random Variables and Symmetry25
2.3Conditioning28
2.4Moment Transforms30
2.5Tail Probability Methods36
2.6Moments of Reciprocals and Ratios38
2.7Reduction of Degree40
2.8Spherical Surface Measure41
2.9Dirichlet Distribution44
2.10Problems46
3Convexity, Optimization, and Inequalities49
3.1Introduction49
3.2Convex Functions50
3.3Minimization of Convex Functions54
3.4The MM Algorithm56
3.5Moment Inequalities60
3.6Problems63
4Combinatorics67
4.1Introduction67
4.2Inclusion-Exclusion67
4.3Applications to Order Statistics72
4.4Stirling Numbers74
4.5Application to an Urn Model77
4.6Pigeonhole Principle79
4.7Problems81
5Combinatorial Optimization85
5.1Introduction85
5.2Quick Sort86
5.3Data Compression and Huffman Coding88
5.4Graph Coloring90
5.5Point Sets with Only Acute Angles94
5.6Sperner's Theorem95
5.7Subadditivity and Expectations96
5.8Problems99
6Poisson Processes103
6.1Introduction103
6.2The Poisson Distribution104
6.3Characterization and Construction104
6.4One-Dimensional Processes107
6.5Transmission Tomography110
6.6Mathematical Applications114
6.7Transformations116
6.8Marking and Coloring117
6.9Campbell's Moment Formulas119
6.10Problems121
7Discrete-Time Markov Chains127
7.1Introduction127
7.2Definitions and Elementary Theory127
7.3Examples130
7.4Coupling134
7.5Hitting Probabilities and Hitting Times139
7.6Markov Chain Monte Carlo141
7.6.1The Hastings-Metropolis Algorithm142
7.6.2Gibbs Sampling143
7.6.3Convergence of the Independence Sampler144
7.7Simulated Annealing145
7.8Problems146
8Continuous-Time Markov Chains151
8.1Introduction151
8.2Finite-Time Transition Probabilities151
8.3Derivation of the Backward Equations153
8.4Equilibrium Distributions and Reversibility154
8.5Examples157
8.6Calculation of Matrix Exponentials160
8.7Kendall's Birth-Death-Immigration Process162
8.8Solution of Kendall's Equation166
8.9Problems168
9Branching Processes175
9.1Introduction175
9.2Examples of Branching Processes176
9.3Elementary Theory177
9.4Extinction179
9.5Immigration183
9.6Multitype Branching Processes187
9.7Viral Reproduction in HIV188
9.8Problems191
10Martingales199
10.1Introduction199
10.2Definition and Examples199
10.3Martingale Convergence203
10.4Optional Stopping206
10.5Large Deviation Bounds210
10.6Problems215
11Diffusion Processes219
11.1Introduction219
11.2Basic Definitions and Properties220
11.3Examples of Diffusion Processes222
11.4Process Moments229
11.5First Passage Problems231
11.6Equilibrium Distribution235
11.7A Numerical Method for Diffusion Processes238
11.8Application to the Wright-Fisher Process242
11.9Problems244
12Poisson Approximation249
12.1Introduction249
12.2Applications of the Coupling Method250
12.3Applications of the Neighborhood Method254
12.4Proof of the Chen-Stein Estimates257
12.5Problems262
13Number Theory267
13.1Introduction267
13.2Zipf's Distribution and Euler's Theorem268
13.3Dirichlet Products and Mobius Inversion272
13.4Averages of Arithmetic Functions276
13.5The Prime Number Theorem279
13.6Problems282
References285
Index295

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