Probability Distributions on Partially Ordered Sets and Positive Semigroups

Description

Posets and Positive Semigroups......Page 4 Algebra......Page 5 Topology......Page 11 Measure......Page 13 Operators and Cumulative Functions......Page 15 Lower and upper operators......Page 16 Cumulative functions......Page 18 Convolution......Page 19 Möbius inversion......Page 20 Isomorphism......Page 21 Sub-semigroups......Page 22 Direct product......Page 26 Ordered groups......Page 29 Lexicographic sums......Page 31 Quotient spaces......Page 33 Distribution functions......Page 36 Residual distributions......Page 42 Expected value......Page 43 Moment generating function......Page 44 Joint distributions......Page 45 Products of independent variables......Page 46 Ladder Variables......Page 48 The point process......Page 50 Definitions and basic properties......Page 51 Entropy......Page 54 Sub-posets......Page 55 Mixtures......Page 57 Special constructions......Page 58 Joint distributions......Page 61 Lexicographic sums......Page 63 Pairs of independent variables......Page 64 Gamma distributions......Page 66 The point process......Page 68 Basic definitions......Page 70 Invariant pairs......Page 71 Basic properties and characterizations......Page 73 Conditional distributions......Page 78 Joint distributions......Page 81 Gamma distributions......Page 84 Quotient spaces......Page 86 III Examples and Applications......Page 89 Gamma distributions and the Point Process......Page 90 Sub-semigroups and quotient spaces......Page 91 The positive semigroup (N, +)......Page 92 Gamma distributions and the point process......Page 93 Sub-semigroups and quotient spaces......Page 95 The positive semigroup......Page 97 Exponential distributions......Page 98 The positive semigroup ((0,1], )......Page 99 The positive semigroup ([1, ), )......Page 100 An application to Brownian functionals......Page 101 Applications to reliability......Page 102 More examples......Page 104 Strong aging properties......Page 105 Definitions......Page 107 Exponential distributions......Page 108 Constant rate distributions......Page 112 Moments......Page 113 Gamma distributions......Page 114 Constant rate distributions......Page 116 Positive semigroups......Page 119 Upper Probability Functions......Page 121 Rate Functions......Page 125 Constant rate distributions......Page 126 Gamma distributions and the point process......Page 127 Relation to Markov Chains......Page 128 Definitions......Page 129 Exponential distributions......Page 132 Gamma distribution......Page 133 Compound Poisson distributions......Page 134 Positive Semigroups with Two Generators......Page 135 Preliminaries......Page 136 The positive semigroup......Page 137 The sub-semigroups......Page 141 Exponential distributions on Tk......Page 143 Almost exponential distributions on S......Page 148 Constant rate distributions on S......Page 154 The cardinality of a constant rate variable......Page 156 Constant rate distirbutions on S, continued......Page 159 Definitions......Page 160 Exponential distributions......Page 161 Definitions......Page 162 Exponential distributions......Page 163 Discrete positive semigroups......Page 164 Equivalence relations and partitions......Page 165