1 A Short Probability Review ................................... 1
1 Sample Space and Random Variables ......................... 1
1.1 Sample Space and Probability Axioms .................. 1
1.2 Discrete Random Variables ............................ 2
2 Independence .............................................. 4
2.1 Binomial Random Variables ............................ 5
2.2 Geometric Random Variables ........................... 6
2.3 A Sum of Poisson Random Variables .................... 7
3 Conditioning .............................................. 8
3.1 Thinning a Poisson Distribution ...................... 9
4 Generating Functions ..................................... 10
4.1 Sum of Poisson Random Variables (Again) ............. 12
4.2 Thinning a Poisson Distribution (Again) ............. 13
Problems ................................................. 13
References ............................................... 15
2 Discrete Time Branching Process ............................. 17
1 The Model ................................................ 17
Problems ................................................. 21
2 The Probability of a Mutation in a Branching Process ..... 24
2.1 An Equation for the Total Progeny Distribution ...... 24
2.2 The Total Progeny Distribution in a Particular
Case ................................................ 25
2.3 The Probability of a Mutation ....................... 28
2.4 Application: The Probability of Drug Resistance ..... 29
2.5 Application: Cancer Risk ............................ 32
2.6 The Total Progeny May Be Infinite ................... 34
Problems ................................................. 36
3 Proof of Theorem 1.1 ..................................... 39
Problems ................................................. 44
Notes .................................................... 45
References ............................................... 46
3 The Simple Symmetric Random Walk ............................ 47
1 Graphical Representation ................................. 48
2 Returns to 0 ............................................. 52
3 Recurrence ............................................... 56
4 Last Return to the Origin ................................ 60
Problems ................................................. 62
Notes .................................................... 65
References ............................................... 65
4 Asymmetric and Higher Dimension Random Walks ................ 67
1 Transience of Asymmetric Random Walks .................... 67
2 Random Walks in Higher Dimensions ........................ 68
2.1 The Two Dimensional Walk ............................ 68
2.2 The Three Dimensional Walk .......................... 69
3 The Ruin Problem ......................................... 71
3.1 The Greedy Gambler .................................. 74
3.2 Random Walk Interpretation .......................... 75
3.3 Duration of the Game ................................ 75
Problems ................................................. 77
5 Discrete Time Markov Chains ................................. 81
1 Classification of States ................................. 81
1.1 Decomposition of the Chain .......................... 82
1.2 A Recurrence Criterion .............................. 84
1.3 Finite Markov Chains ................................ 88
Problems ................................................. 90
2 Birth and Death Chains ................................... 91
2.1 The Coupling Technique .............................. 96
2.2 An Application: Quorum Sensing ...................... 98
Problems ................................................ 100
Notes ................................................... 103
References .............................................. 103
6 Stationary Distributions for Discrete Time Markov Chains ... 105
1 Convergence to a Stationary Distribution ................ 105
1.1 Convergence and Positive Recurrence ................ 105
1.2 Stationary Distributions ........................... 109
1.3 The Finite Case .................................... 113
Problems ................................................ 115
2 Examples and Applications ............................... 117
2.1 Reversibility ...................................... 117
2.2 Birth and Death Chains ............................. 118
2.3 The Simple Random Walk on the Half Line ............ 120
2.4 The Ehrenfest Chain ................................ 122
2.5 The First Appearance of a Pattern .................. 125
Problems ................................................ 126
Notes ................................................... 128
References .............................................. 129
7 The Poisson Process ........................................ 131
1 The Exponential Distribution ............................ 131
Problems ................................................ 134
2 The Poisson Process ..................................... 135
2.1 Application: Influenza Pandemics .................. 145
Problems ................................................ 146
Notes ................................................... 148
References .............................................. 149
8 Continuous Time Branching Processes ........................ 151
1 A Continuous Time Binary Branching Process .............. 151
Problems ................................................ 153
2 A Model for Immune Response ............................. 155
2.1 The Tree of Types .................................. 156
Problems ................................................ 160
3 A Model for Virus Survival .............................. 162
Problems ................................................ 168
4 A Model for Bacterial Persistence ....................... 169
Problems ................................................ 172
Notes ................................................... 173
References .............................................. 173
9 Continuous Time Birth and Death Chains ..................... 175
1 The Kolmogorov Differential Equations ................... 175
1.1 The Pure Birth Processes ........................... 178
1.2 The Yule Process ................................... 180
1.3 The Yule Process with Mutations .................... 181
1.4 Passage Times ...................................... 182
Problems ................................................ 184
2 Limiting Probabilities .................................. 186
Problems ................................................ 193
Notes ................................................... 195
References .............................................. 195
10 Percolation ................................................ 197
1 Percolation on the Lattice .............................. 197
Problems ................................................ 202
2 Further Properties of Percolation ....................... 205
2.1 Continuity of the Percolation Probability .......... 205
2.2 The Subcritical Phase .............................. 208
Problems ................................................ 212
3 Percolation On a Tree and Two Critical Exponents ........ 214
Problems ................................................ 216
Notes ................................................... 217
References .............................................. 218
11 A Cellular Automaton ....................................... 219
1 The Model ............................................... 219
Problems ................................................ 222
2 A Renormalization Argument .............................. 224
Problems ................................................ 228
Notes ................................................... 229
References .............................................. 229
12 A Branching Random Walk .................................... 231
1 The Model ............................................... 231
1.1 The Branching Random Walk on the Line .............. 233
1.2 The Branching Random Walk on a Tree ................ 235
1.3 Proof of Theorem 1.1 ............................... 237
Problems ................................................ 242
2 Continuity of the Phase Transitions ..................... 243
2.1 The First Phase Transition Is Continuous ........... 243
2.2 The Second Phase Transition Is Discontinuous ....... 246
Problems ................................................ 249
Notes ................................................... 249
References .............................................. 249
13 The Contact Process on a Homogeneous Tree .................. 251
1 The Two Phase Transitions ............................... 251
Problems ................................................ 253
2 Characterization of the First Phase Transition .......... 254
Problems ................................................ 255
Notes ................................................... 256
References .............................................. 256
A A Little More Probability .................................. 257
1 Probability Space ....................................... 257
Problems ................................................ 260
2 Borel-Cantelli Lemma .................................... 261
2.1 Infinite Products .................................. 263
Problems ................................................ 264
Notes ................................................... 265
References .............................................. 265
Index ......................................................... 267
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