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MIT RES.6-012 Introduction to Probability, Spring 2018

MIT RES.6-012 Introduction to Probability, Spring 2018

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L01.6 More Properties of Probabilities

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L01.6 More Properties of Probabilities

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MIT RES.6-012 Introduction to Probability, Spring 2018

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  1. 1 L01.1 Lecture Overview
  2. 2 L01.2 Sample Space
  3. 3 L01.3 Sample Space Examples
  4. 4 L01.4 Probability Axioms
  5. 5 L01.5 Simple Properties of Probabilities
  6. 6 L01.6 More Properties of Probabilities
  7. 7 L01.7 A Discrete Example
  8. 8 L01.8 A Continuous Example
  9. 9 L01.9 Countable Additivity
  10. 10 L01.10 Interpretations & Uses of Probabilities
  11. 11 S01.0 Mathematical Background Overview
  12. 12 S01.1 Sets
  13. 13 S01.2 De Morgan's Laws
  14. 14 S01.3 Sequences and their Limits
  15. 15 S01.4 When Does a Sequence Converge
  16. 16 S01.5 Infinite Series
  17. 17 S01.6 The Geometric Series
  18. 18 S01.7 About the Order of Summation in Series with Multiple Indices
  19. 19 S01.8 Countable and Uncountable Sets
  20. 20 S01.9 Proof That a Set of Real Numbers is Uncountable
  21. 21 S01.10 Bonferroni's Inequality
  22. 22 L02.1 Lecture Overview
  23. 23 L02.2 Conditional Probabilities
  24. 24 L02.3 A Die Roll Example
  25. 25 L02.4 Conditional Probabilities Obey the Same Axioms
  26. 26 L02.5 A Radar Example and Three Basic Tools
  27. 27 L02.6 The Multiplication Rule
  28. 28 L02.7 Total Probability Theorem
  29. 29 L02.8 Bayes' Rule
  30. 30 L03.1 Lecture Overview
  31. 31 L03.2 A Coin Tossing Example
  32. 32 L03.3 Independence of Two Events
  33. 33 L03.4 Independence of Event Complements
  34. 34 L03.5 Conditional Independence
  35. 35 L03.6 Independence Versus Conditional Independence
  36. 36 L03.7 Independence of a Collection of Events
  37. 37 L03.8 Independence Versus Pairwise Independence
  38. 38 L03.9 Reliability
  39. 39 L03.10 The King's Sibling
  40. 40 L04.1 Lecture Overview
  41. 41 L04.2 The Counting Principle
  42. 42 L04.3 Die Roll Example
  43. 43 L04.4 Combinations
  44. 44 L04.5 Binomial Probabilities
  45. 45 L04.6 A Coin Tossing Example
  46. 46 L04.7 Partitions
  47. 47 L04.8 Each Person Gets An Ace
  48. 48 L04.9 Multinomial Probabilities
  49. 49 L05.1 Lecture Overview
  50. 50 L05.2 Definition of Random Variables
  51. 51 L05.3 Probability Mass Functions
  52. 52 L05.4 Bernoulli & Indicator Random Variables
  53. 53 L05.5 Uniform Random Variables
  54. 54 L05.6 Binomial Random Variables
  55. 55 L05.7 Geometric Random Variables
  56. 56 L05.8 Expectation
  57. 57 L05.9 Elementary Properties of Expectation
  58. 58 L05.10 The Expected Value Rule
  59. 59 L05.11 Linearity of Expectations
  60. 60 S05.1 Supplement: Functions
  61. 61 L06.1 Lecture Overview
  62. 62 L06.2 Variance
  63. 63 L06.3 The Variance of the Bernoulli & The Uniform
  64. 64 L06.4 Conditional PMFs & Expectations Given an Event
  65. 65 L06.5 Total Expectation Theorem
  66. 66 L06.6 Geometric PMF Memorylessness & Expectation
  67. 67 L06.7 Joint PMFs and the Expected Value Rule
  68. 68 L06.8 Linearity of Expectations & The Mean of the Binomial
  69. 69 L07.1 Lecture Overview
  70. 70 L07.2 Conditional PMFs
  71. 71 L07.3 Conditional Expectation & the Total Expectation Theorem
  72. 72 L07.4 Independence of Random Variables
  73. 73 L07.5 Example
  74. 74 L07.6 Independence & Expectations
  75. 75 L07.7 Independence, Variances & the Binomial Variance
  76. 76 L07.8 The Hat Problem
  77. 77 S07.1 The Inclusion-Exclusion Formula
  78. 78 S07.2 The Variance of the Geometric
  79. 79 S07.3 Independence of Random Variables Versus Independence of Events
  80. 80 L08.1 Lecture Overview
  81. 81 L08.2 Probability Density Functions
  82. 82 L08.3 Uniform & Piecewise Constant PDFs
  83. 83 L08.4 Means & Variances
  84. 84 L08.5 Mean & Variance of the Uniform
  85. 85 L08.6 Exponential Random Variables
  86. 86 L08.7 Cumulative Distribution Functions
  87. 87 L08.8 Normal Random Variables
  88. 88 L08.9 Calculation of Normal Probabilities
  89. 89 L09.1 Lecture Overview
  90. 90 L09.2 Conditioning A Continuous Random Variable on an Event
  91. 91 L09.3 Conditioning Example
  92. 92 L09.4 Memorylessness of the Exponential PDF
  93. 93 L09.5 Total Probability & Expectation Theorems
  94. 94 L09.6 Mixed Random Variables
  95. 95 L09.7 Joint PDFs
  96. 96 L09.8 From The Joint to the Marginal
  97. 97 L09.9 Continuous Analogs of Various Properties
  98. 98 L09.10 Joint CDFs
  99. 99 S09.1 Buffon's Needle & Monte Carlo Simulation
  100. 100 L10.1 Lecture Overview
  101. 101 L10.2 Conditional PDFs
  102. 102 L10.3 Comments on Conditional PDFs
  103. 103 L10.4 Total Probability & Total Expectation Theorems
  104. 104 L10.5 Independence
  105. 105 L10.6 Stick-Breaking Example
  106. 106 L10.7 Independent Normals
  107. 107 L10.8 Bayes Rule Variations
  108. 108 L10.9 Mixed Bayes Rule
  109. 109 L10.10 Detection of a Binary Signal
  110. 110 L10.11 Inference of the Bias of a Coin
  111. 111 L11.1 Lecture Overview
  112. 112 L11.2 The PMF of a Function of a Discrete Random Variable
  113. 113 L11.3 A Linear Function of a Continuous Random Variable
  114. 114 L11.4 A Linear Function of a Normal Random Variable
  115. 115 L11.5 The PDF of a General Function
  116. 116 L11.6 The Monotonic Case
  117. 117 L11.7 The Intuition for the Monotonic Case
  118. 118 L11.8 A Nonmonotonic Example
  119. 119 L11.9 The PDF of a Function of Multiple Random Variables
  120. 120 S11.1 Simulation
  121. 121 L12.1 Lecture Overview
  122. 122 L12.2 The Sum of Independent Discrete Random Variables
  123. 123 L12.3 The Sum of Independent Continuous Random Variables
  124. 124 L12.4 The Sum of Independent Normal Random Variables
  125. 125 L12.5 Covariance
  126. 126 L12.6 Covariance Properties
  127. 127 L12.7 The Variance of the Sum of Random Variables
  128. 128 L12.8 The Correlation Coefficient
  129. 129 L12.9 Proof of Key Properties of the Correlation Coefficient
  130. 130 L12.10 Interpreting the Correlation Coefficient
  131. 131 L12.11 Correlations Matter
  132. 132 L13.1 Lecture Overview
  133. 133 L13.2 Conditional Expectation as a Random Variable
  134. 134 L13.3 The Law of Iterated Expectations
  135. 135 L13.4 Stick-Breaking Revisited
  136. 136 L13.5 Forecast Revisions
  137. 137 L13.6 The Conditional Variance
  138. 138 L13.7 Derivation of the Law of Total Variance
  139. 139 L13.8 A Simple Example
  140. 140 L13.9 Section Means and Variances
  141. 141 L13.10 Mean of the Sum of a Random Number of Random Variables
  142. 142 L13.11 Variance of the Sum of a Random Number of Random Variables
  143. 143 S13.1 Conditional Expectation Properties
  144. 144 L14.1 Lecture Overview
  145. 145 L14.2 Overview of Some Application Domains
  146. 146 L14.3 Types of Inference Problems
  147. 147 L14.4 The Bayesian Inference Framework
  148. 148 L14.5 Discrete Parameter, Discrete Observation
  149. 149 L14.6 Discrete Parameter, Continuous Observation
  150. 150 L14.7 Continuous Parameter, Continuous Observation
  151. 151 L14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution
  152. 152 L14.9 Inferring the Unknown Bias of a Coin - Point Estimates
  153. 153 L14.10 Summary
  154. 154 S14.1 The Beta Formula
  155. 155 L15.1 Lecture Overview
  156. 156 L15.2 Recognizing Normal PDFs
  157. 157 L15.3 Estimating a Normal Random Variable in the Presence of Additive Noise
  158. 158 L15.4 The Case of Multiple Observations
  159. 159 L15.5 The Mean Squared Error
  160. 160 L15.6 Multiple Parameters; Trajectory Estimation
  161. 161 L15.7 Linear Normal Models
  162. 162 L15.8 Trajectory Estimation Illustration
  163. 163 L16.1 Lecture Overview
  164. 164 L16.2 LMS Estimation in the Absence of Observations
  165. 165 L16.3 LMS Estimation of One Random Variable Based on Another
  166. 166 L16.4 LMS Performance Evaluation
  167. 167 L16.5 Example: The LMS Estimate
  168. 168 L16.6 Example Continued: LMS Performance Evaluation
  169. 169 L16.7 LMS Estimation with Multiple Observations or Unknowns
  170. 170 L16.8 Properties of the LMS Estimation Error
  171. 171 L17.1 Lecture Overview
  172. 172 L17.2 LLMS Formulation
  173. 173 L17.3 Solution to the LLMS Problem
  174. 174 L17.4 Remarks on the LLMS Solution and on the Error Variance
  175. 175 L17.5 LLMS Example
  176. 176 L17.6 LLMS for Inferring the Parameter of a Coin
  177. 177 L17.7 LLMS with Multiple Observations
  178. 178 L17.8 The Simplest LLMS Example with Multiple Observations
  179. 179 L17.9 The Representation of the Data Matters in LLMS
  180. 180 L18.1 Lecture Overview
  181. 181 L18.2 The Markov Inequality
  182. 182 L18.3 The Chebyshev Inequality
  183. 183 L18.4 The Weak Law of Large Numbers
  184. 184 L18.5 Polling
  185. 185 L18.6 Convergence in Probability
  186. 186 L18.7 Convergence in Probability Examples
  187. 187 L18.8 Related Topics
  188. 188 S18.1 Convergence in Probability of the Sum of Two Random Variables
  189. 189 S18.2 Jensen's Inequality
  190. 190 S18.3 Hoeffding's Inequality
  191. 191 L19.1 Lecture Overview
  192. 192 L19.2 The Central Limit Theorem
  193. 193 L19.3 Discussion of the CLT
  194. 194 L19.4 Illustration of the CLT
  195. 195 L19.5 CLT Examples
  196. 196 L19.6 Normal Approximation to the Binomial
  197. 197 L19.7 Polling Revisited
  198. 198 L20.1 Lecture Overview
  199. 199 L20.2 Overview of the Classical Statistical Framework
  200. 200 L20.3 The Sample Mean and Some Terminology
  201. 201 L20.4 On the Mean Squared Error of an Estimator
  202. 202 L20.5 Confidence Intervals
  203. 203 L20.6 Confidence Intervals for the Estimation of the Mean
  204. 204 L20.7 Confidence Intervals for the Mean, When the Variance is Unknown
  205. 205 L20.8 Other Natural Estimators
  206. 206 L20.9 Maximum Likelihood Estimation
  207. 207 L20.10 Maximum Likelihood Estimation Examples
  208. 208 L21.1 Lecture Overview
  209. 209 L21.2 The Bernoulli Process
  210. 210 L21.3 Stochastic Processes
  211. 211 L21.4 Review of Known Properties of the Bernoulli Process
  212. 212 L21.5 The Fresh Start Property
  213. 213 L21.6 Example: The Distribution of a Busy Period
  214. 214 L21.7 The Time of the K-th Arrival
  215. 215 L21.8 Merging of Bernoulli Processes
  216. 216 L21.9 Splitting a Bernoulli Process
  217. 217 L21.10 The Poisson Approximation to the Binomial
  218. 218 L22.1 Lecture Overview
  219. 219 L22.2 Definition of the Poisson Process
  220. 220 L22.3 Applications of the Poisson Process
  221. 221 L22.4 The Poisson PMF for the Number of Arrivals
  222. 222 L22.5 The Mean and Variance of the Number of Arrivals
  223. 223 L22.6 A Simple Example
  224. 224 L22.7 Time of the K-th Arrival
  225. 225 L22.8 The Fresh Start Property and Its Implications
  226. 226 L22.9 Summary of Results
  227. 227 L22.10 An Example
  228. 228 L23.1 Lecture Overview
  229. 229 L23.2 The Sum of Independent Poisson Random Variables
  230. 230 L23.3 Merging Independent Poisson Processes
  231. 231 L23.4 Where is an Arrival of the Merged Process Coming From?
  232. 232 L23.5 The Time Until the First (or last) Lightbulb Burns Out
  233. 233 L23.6 Splitting a Poisson Process
  234. 234 L23.7 Random Incidence in the Poisson Process
  235. 235 L23.8 Random Incidence in a Non-Poisson Process
  236. 236 L23.9 Different Sampling Methods can Give Different Results
  237. 237 S23.1 Poisson Versus Normal Approximations to the Binomial
  238. 238 S23.2 Poisson Arrivals During an Exponential Interval
  239. 239 L24.1 Lecture Overview
  240. 240 L24.2 Introduction to Markov Processes
  241. 241 L24.3 Checkout Counter Example
  242. 242 L24.4 Discrete-Time Finite-State Markov Chains
  243. 243 L24.5 N-Step Transition Probabilities
  244. 244 L24.6 A Numerical Example - Part I
  245. 245 L24.7 Generic Convergence Questions
  246. 246 L24.8 Recurrent and Transient States
  247. 247 L25.1 Brief Introduction (RES.6-012 Introduction to Probability)
  248. 248 L25.2 Lecture Overview
  249. 249 L25.3 Markov Chain Review
  250. 250 L25.4 The Probability of a Path
  251. 251 L25.5 Recurrent and Transient States: Review
  252. 252 L25.6 Periodic States
  253. 253 L25.7 Steady-State Probabilities and Convergence
  254. 254 L25.8 A Numerical Example - Part II
  255. 255 L25.9 Visit Frequency Interpretation of Steady-State Probabilities
  256. 256 L25.10 Birth-Death Processes - Part I
  257. 257 L25.11 Birth-Death Processes - Part II
  258. 258 L26.1 Brief Introduction (RES.6-012 Introduction to Probability)
  259. 259 L26.2 Lecture Overview
  260. 260 L26.3 Review of Steady-State Behavior
  261. 261 L26.4 A Numerical Example - Part III
  262. 262 L26.5 Design of a Phone System
  263. 263 L26.6 Absorption Probabilities
  264. 264 L26.7 Expected Time to Absorption
  265. 265 L26.8 Mean First Passage Time
  266. 266 L26.9 Gambler's Ruin

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