Fundamental Probability Book

Preface     xi
A note to the student (and instructor)     xvi
A note to the instructor (and student)     xviii
Acknowledgements     xxi
Introduction     1
Basic Probability     7
Combinatorics     9
Basic counting     9
Generalized binomial coefficients     13
Combinatoric identities and the use of induction     15
The binomial and multinomial theorems     18
The binomial theorem     18
An extension of the binomial theorem     23
The multinomial theorem     27
The gamma and beta functions     28
The gamma function     28
The beta function     31
Problems     36
Probability spaces and counting     43
Introducing counting and occupancy problems     43
Probability spaces     47
Introduction     47
Definitions     49
Properties     58
Basic properties     58
Advanced properties     59
A theoretical property     67
Problems     68
Symmetric spaces and conditioning     73
Applications withsymmetric probability spaces     73
Conditional probability and independence     85
Total probability and Bayes' rule     87
Extending the law of total probability     93
Statistical paradoxes and fallacies     96
The problem of the points     97
Three solutions     97
Further gambling problems     99
Some historical references     100
Problems     101
Discrete Random Variables     111
Univariate random variables     113
Definitions and properties     113
Basic definitions and properties     113
Further definitions and properties     117
Discrete sampling schemes     120
Bernoulli and binomial     121
Hypergeometric     123
Geometric and negative binomial     125
Inverse hypergeometric     128
Poisson approximations     130
Occupancy distributions     133
Transformations     140
Moments     141
Expected value of X     141
Higher-order moments     143
Jensen's inequality     151
Poisson processes     154
Problems      156
Multivariate random variables     165
Multivariate density and distribution     165
Joint cumulative distribution functions     166
Joint probability mass and density functions     168
Fundamental properties of multivariate random variables     171
Marginal distributions     171
Independence     173
Exchangeability     174
Transformations     175
Moments     176
Discrete sampling schemes     182
Multinomial     182
Multivariate hypergeometric     188
Multivariate negative binomial     190
Multivariate inverse hypergeometric     192
Problems     194
Sums of random variables     197
Mean and variance     197
Use of exchangeable Bernoulli random variables     199
Examples with birthdays     202
Runs distributions     206
Random variable decomposition     218
Binomial, negative binomial and Poisson     218
Hypergeometric     220
Inverse hypergeometric     222
General linear combination of two random variables     227
Problems      232
Continuous Random Variables     237
Continuous univariate random variables     239
Most prominent distributions     239
Other popular distributions     263
Univariate transformations     269
Examples of one-to-one transformations     271
Many-to-one transformations     273
The probability integral transform     275
Simulation     276
Kernel density estimation     277
Problems     278
Joint and conditional random variables     285
Review of basic concepts     285
Conditional distributions     290
Discrete case     291
Continuous case     292
Conditional moments     304
Expected shortfall     310
Independence     311
Computing probabilities via conditioning     312
Problems     317
Multivariate transformations     323
Basic transformation     323
The t and F distributions     329
Further aspects and important transformations     333
Problems     339
Appendices     343
Calculus review     343
Recommended reading      343
Sets, functions and fundamental inequalities     345
Univariate calculus     350
Limits and continuity     351
Differentiation     352
Integration     364
Series     382
Multivariate calculus     413
Neighborhoods and open sets     413
Sequences, limits and continuity     414
Differentiation     416
Integration     425
Notation tables     435
Distribution tables     441
References     451
Index     461