Probability, Mathematical Statistics, and Stochastic Processes

by Kyle Siegrist

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This text is devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects. The site consists of an integrated set of components that includes expository text, interactive web apps, data sets, biographical sketches, and an object library.

Description from LibreTexts.

Publisher
University of Alabama in huntsville
Subjects
Statistics, Probability Theory
Licence
CC BY 2.0, as LibreTexts states it
Who can read
Tweens profiles and older, judged from its title and subjects

Contents

LibreTexts writes equations in TeX, and here they show as TeX; the PDF has them typeset.

  1. Front Matter
  2. TitlePage
  3. InfoPage
  4. Introduction
  5. Table of Contents
  6. Licensing
  7. Table of Contents
  8. Object Library
  9. Credits
  10. Sources and Resources
  11. 1: Foundations
  12. 1.1: Sets
  13. 1.2: Functions
  14. 1.3: Relations
  15. 1.4: Partial Orders
  16. 1.5: Equivalence Relations
  17. 1.6: Cardinality
  18. 1.7: Counting Measure
  19. 1.8: Combinatorial Structures
  20. 1.9: Topological Spaces
  21. 1.10: Metric Spaces
  22. 1.11: Measurable Spaces
  23. 1.12: Special Set Structures
  24. 2: Probability Spaces
  25. 2.1: Random Experiments
  26. 2.2: Events and Random Variables
  27. 2.3: Probability Measures
  28. 2.4: Conditional Probability
  29. 2.5: Independence
  30. 2.6: Convergence
  31. 2.7: Measure Spaces
  32. 2.8: Existence and Uniqueness
  33. 2.9: Probability Spaces Revisited
  34. 2.10: Stochastic Processes
  35. 2.11: Filtrations and Stopping Times
  36. 3: Distributions
  37. 3.1: Discrete Distributions
  38. 3.2: Continuous Distributions
  39. 3.3: Mixed Distributions
  40. 3.4: Joint Distributions
  41. 3.5: Conditional Distributions
  42. 3.6: Distribution and Quantile Functions
  43. 3.7: Transformations of Random Variables
  44. 3.8: Convergence in Distribution
  45. 3.9: General Distribution Functions
  46. 3.10: The Integral With Respect to a Measure
  47. 3.11: Properties of the Integral
  48. 3.12: General Measures
  49. 3.13: Absolute Continuity and Density Functions
  50. 3.14: Function Spaces
  51. 4: Expected Value
  52. 4.1: Definitions and Basic Properties
  53. 4.2: Additional Properties
  54. 4.3: Variance
  55. 4.4: Skewness and Kurtosis
  56. 4.5: Covariance and Correlation
  57. 4.6: Generating Functions
  58. 4.7: Conditional Expected Value
  59. 4.8: Expected Value and Covariance Matrices
  60. 4.9: Expected Value as an Integral
  61. 4.10: Conditional Expected Value Revisited
  62. 4.11: Vector Spaces of Random Variables
  63. 4.12: Uniformly Integrable Variables
  64. 4.13: Kernels and Operators
  65. 5: Special Distributions
  66. 5.1: Location-Scale Families
  67. 5.2: General Exponential Families
  68. 5.3: Stable Distributions
  69. 5.4: Infinitely Divisible Distributions
  70. 5.5: Power Series Distributions
  71. 5.6: The Normal Distribution
  72. 5.7: The Multivariate Normal Distribution
  73. 5.8: The Gamma Distribution
  74. 5.9: Chi-Square and Related Distribution
  75. 5.10: The Student t Distribution
  76. 5.11: The F Distribution
  77. 5.12: The Lognormal Distribution
  78. 5.13: The Folded Normal Distribution
  79. 5.14: The Rayleigh Distribution
  80. 5.15: The Maxwell Distribution
  81. 5.16: The Lévy Distribution
  82. 5.17: The Beta Distribution
  83. 5.18: The Beta Prime Distribution
  84. 5.19: The Arcsine Distribution
  85. 5.20: General Uniform Distributions
  86. 5.21: The Uniform Distribution on an Interval
  87. 5.22: Discrete Uniform Distributions
  88. 5.23: The Semicircle Distribution
  89. 5.24: The Triangle Distribution
  90. 5.25: The Irwin-Hall Distribution
  91. 5.26: The U-Power Distribution
  92. 5.27: The Sine Distribution
  93. 5.28: The Laplace Distribution
  94. 5.29: The Logistic Distribution
  95. 5.30: The Extreme Value Distribution
  96. 5.31: The Hyperbolic Secant Distribution
  97. 5.32: The Cauchy Distribution
  98. 5.33: The Exponential-Logarithmic Distribution
  99. 5.34: The Gompertz Distribution
  100. 5.35: The Log-Logistic Distribution
  101. 5.36: The Pareto Distribution
  102. 5.37: The Wald Distribution
  103. 5.38: The Weibull Distribution
  104. 5.39: Benford's Law
  105. 5.40: The Zeta Distribution
  106. 5.41: The Logarithmic Series Distribution
  107. 6: Random Samples
  108. 6.1: Introduction
  109. 6.2: The Sample Mean
  110. 6.3: The Law of Large Numbers
  111. 6.4: The Central Limit Theorem
  112. 6.5: The Sample Variance
  113. 6.6: Order Statistics
  114. 6.7: Sample Correlation and Regression
  115. 6.8: Special Properties of Normal Samples
  116. 7: Point Estimation
  117. 7.1: Estimators
  118. 7.2: The Method of Moments
  119. 7.3: Maximum Likelihood
  120. 7.4: Bayesian Estimation
  121. 7.5: Best Unbiased Estimators
  122. 7.6: Sufficient, Complete and Ancillary Statistics
  123. 8: Set Estimation
  124. 8.1: Introduction to Set Estimation
  125. 8.2: Estimation the Normal Model
  126. 8.3: Estimation in the Bernoulli Model
  127. 8.4: Estimation in the Two-Sample Normal Model
  128. 8.5: Bayesian Set Estimation
  129. 9: Hypothesis Testing
  130. 9.1: Introduction to Hypothesis Testing
  131. 9.2: Tests in the Normal Model
  132. 9.3: Tests in the Bernoulli Model
  133. 9.4: Tests in the Two-Sample Normal Model
  134. 9.5: Likelihood Ratio Tests
  135. 9.6: Chi-Square Tests
  136. 10: Geometric Models
  137. 10.1: Buffon's Problems
  138. 10.2: Bertrand's Paradox
  139. 10.3: Random Triangles
  140. 11: Bernoulli Trials
  141. 11.1: Introduction to Bernoulli Trials
  142. 11.2: The Binomial Distribution
  143. 11.3: The Geometric Distribution
  144. 11.4: The Negative Binomial Distribution
  145. 11.5: The Multinomial Distribution
  146. 11.6: The Simple Random Walk
  147. 11.7: The Beta-Bernoulli Process
  148. 12: Finite Sampling Models
  149. 12.1: Introduction to Finite Sampling Models
  150. 12.2: The Hypergeometric Distribution
  151. 12.3: The Multivariate Hypergeometric Distribution
  152. 12.4: Order Statistics
  153. 12.5: The Matching Problem
  154. 12.6: The Birthday Problem
  155. 12.7: The Coupon Collector Problem
  156. 12.8: Pólya's Urn Process
  157. 12.9: The Secretary Problem
  158. 13: Games of Chance
  159. 13.1: Introduction to Games of Chance
  160. 13.2: Poker
  161. 13.3: Simple Dice Games
  162. 13.4: Craps
  163. 13.5: Roulette
  164. 13.6: The Monty Hall Problem
  165. 13.7: Lotteries
  166. 13.8: The Red and Black Game
  167. 13.9: Timid Play
  168. 13.10: Bold Play
  169. 13.11: Optimal Strategies
  170. 14: The Poisson Process
  171. 14.1: Introduction to the Poisson Process
  172. 14.2: The Exponential Distribution
  173. 14.3: The Gamma Distribution
  174. 14.4: The Poisson Distribution
  175. 14.5: Thinning and Superpositon
  176. 14.6: Non-homogeneous Poisson Processes
  177. 14.7: Compound Poisson Processes
  178. 14.8: Poisson Processes on General Spaces
  179. 15: Renewal Processes
  180. 15.1: Introduction
  181. 15.2: Renewal Equations
  182. 15.3: Renewal Limit Theorems
  183. 15.4: Delayed Renewal Processes
  184. 15.5: Alternating Renewal Processes
  185. 15.6: Renewal Reward Processes
  186. 16: Markov Processes
  187. 16.1: Introduction to Markov Processes
  188. 16.2: Potentials and Generators for General Markov Processes
  189. 16.3: Introduction to Discrete-Time Chains
  190. 16.4: Transience and Recurrence for Discrete-Time Chains
  191. 16.5: Periodicity of Discrete-Time Chains
  192. 16.6: Stationary and Limiting Distributions of Discrete-Time Chains
  193. 16.7: Time Reversal in Discrete-Time Chains
  194. 16.8: The Ehrenfest Chains
  195. 16.9: The Bernoulli-Laplace Chain
  196. 16.10: Discrete-Time Reliability Chains
  197. 16.11: Discrete-Time Branching Chain
  198. 16.12: Discrete-Time Queuing Chains
  199. 16.13: Discrete-Time Birth-Death Chains
  200. 16.14: Random Walks on Graphs
  201. 16.15: Introduction to Continuous-Time Markov Chains
  202. 16.16: Transition Matrices and Generators of Continuous-Time Chains
  203. 16.17: Potential Matrices
  204. 16.18: Stationary and Limting Distributions of Continuous-Time Chains
  205. 16.19: Time Reversal in Continuous-Time Chains
  206. 16.20: Chains Subordinate to the Poisson Process
  207. 16.21: Continuous-Time Birth-Death Chains
  208. 16.22: Continuous-Time Queuing Chains
  209. 16.23: Continuous-Time Branching Chains
  210. 17: Martingales
  211. 17.1: Introduction to Martingalges
  212. 17.2: Properties and Constructions
  213. 17.3: Stopping Times
  214. 17.4: Inequalities
  215. 17.5: Convergence
  216. 17.6: Backwards Martingales
  217. 18: Brownian Motion
  218. 18.1: Standard Brownian Motion
  219. 18.2: Brownian Motion with Drift and Scaling
  220. 18.3: The Brownian Bridge
  221. 18.4: Geometric Brownian Motion
  222. Back Matter
  223. Index
  224. Glossary
  225. Detailed Licensing