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L22.8 The Fresh Start Property and Its Implications
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MIT RES.6-012 Introduction to Probability, Spring 2018
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- 1 L01.1 Lecture Overview
- 2 L01.2 Sample Space
- 3 L01.3 Sample Space Examples
- 4 L01.4 Probability Axioms
- 5 L01.5 Simple Properties of Probabilities
- 6 L01.6 More Properties of Probabilities
- 7 L01.7 A Discrete Example
- 8 L01.8 A Continuous Example
- 9 L01.9 Countable Additivity
- 10 L01.10 Interpretations & Uses of Probabilities
- 11 S01.0 Mathematical Background Overview
- 12 S01.1 Sets
- 13 S01.2 De Morgan's Laws
- 14 S01.3 Sequences and their Limits
- 15 S01.4 When Does a Sequence Converge
- 16 S01.5 Infinite Series
- 17 S01.6 The Geometric Series
- 18 S01.7 About the Order of Summation in Series with Multiple Indices
- 19 S01.8 Countable and Uncountable Sets
- 20 S01.9 Proof That a Set of Real Numbers is Uncountable
- 21 S01.10 Bonferroni's Inequality
- 22 L02.1 Lecture Overview
- 23 L02.2 Conditional Probabilities
- 24 L02.3 A Die Roll Example
- 25 L02.4 Conditional Probabilities Obey the Same Axioms
- 26 L02.5 A Radar Example and Three Basic Tools
- 27 L02.6 The Multiplication Rule
- 28 L02.7 Total Probability Theorem
- 29 L02.8 Bayes' Rule
- 30 L03.1 Lecture Overview
- 31 L03.2 A Coin Tossing Example
- 32 L03.3 Independence of Two Events
- 33 L03.4 Independence of Event Complements
- 34 L03.5 Conditional Independence
- 35 L03.6 Independence Versus Conditional Independence
- 36 L03.7 Independence of a Collection of Events
- 37 L03.8 Independence Versus Pairwise Independence
- 38 L03.9 Reliability
- 39 L03.10 The King's Sibling
- 40 L04.1 Lecture Overview
- 41 L04.2 The Counting Principle
- 42 L04.3 Die Roll Example
- 43 L04.4 Combinations
- 44 L04.5 Binomial Probabilities
- 45 L04.6 A Coin Tossing Example
- 46 L04.7 Partitions
- 47 L04.8 Each Person Gets An Ace
- 48 L04.9 Multinomial Probabilities
- 49 L05.1 Lecture Overview
- 50 L05.2 Definition of Random Variables
- 51 L05.3 Probability Mass Functions
- 52 L05.4 Bernoulli & Indicator Random Variables
- 53 L05.5 Uniform Random Variables
- 54 L05.6 Binomial Random Variables
- 55 L05.7 Geometric Random Variables
- 56 L05.8 Expectation
- 57 L05.9 Elementary Properties of Expectation
- 58 L05.10 The Expected Value Rule
- 59 L05.11 Linearity of Expectations
- 60 S05.1 Supplement: Functions
- 61 L06.1 Lecture Overview
- 62 L06.2 Variance
- 63 L06.3 The Variance of the Bernoulli & The Uniform
- 64 L06.4 Conditional PMFs & Expectations Given an Event
- 65 L06.5 Total Expectation Theorem
- 66 L06.6 Geometric PMF Memorylessness & Expectation
- 67 L06.7 Joint PMFs and the Expected Value Rule
- 68 L06.8 Linearity of Expectations & The Mean of the Binomial
- 69 L07.1 Lecture Overview
- 70 L07.2 Conditional PMFs
- 71 L07.3 Conditional Expectation & the Total Expectation Theorem
- 72 L07.4 Independence of Random Variables
- 73 L07.5 Example
- 74 L07.6 Independence & Expectations
- 75 L07.7 Independence, Variances & the Binomial Variance
- 76 L07.8 The Hat Problem
- 77 S07.1 The Inclusion-Exclusion Formula
- 78 S07.2 The Variance of the Geometric
- 79 S07.3 Independence of Random Variables Versus Independence of Events
- 80 L08.1 Lecture Overview
- 81 L08.2 Probability Density Functions
- 82 L08.3 Uniform & Piecewise Constant PDFs
- 83 L08.4 Means & Variances
- 84 L08.5 Mean & Variance of the Uniform
- 85 L08.6 Exponential Random Variables
- 86 L08.7 Cumulative Distribution Functions
- 87 L08.8 Normal Random Variables
- 88 L08.9 Calculation of Normal Probabilities
- 89 L09.1 Lecture Overview
- 90 L09.2 Conditioning A Continuous Random Variable on an Event
- 91 L09.3 Conditioning Example
- 92 L09.4 Memorylessness of the Exponential PDF
- 93 L09.5 Total Probability & Expectation Theorems
- 94 L09.6 Mixed Random Variables
- 95 L09.7 Joint PDFs
- 96 L09.8 From The Joint to the Marginal
- 97 L09.9 Continuous Analogs of Various Properties
- 98 L09.10 Joint CDFs
- 99 S09.1 Buffon's Needle & Monte Carlo Simulation
- 100 L10.1 Lecture Overview
- 101 L10.2 Conditional PDFs
- 102 L10.3 Comments on Conditional PDFs
- 103 L10.4 Total Probability & Total Expectation Theorems
- 104 L10.5 Independence
- 105 L10.6 Stick-Breaking Example
- 106 L10.7 Independent Normals
- 107 L10.8 Bayes Rule Variations
- 108 L10.9 Mixed Bayes Rule
- 109 L10.10 Detection of a Binary Signal
- 110 L10.11 Inference of the Bias of a Coin
- 111 L11.1 Lecture Overview
- 112 L11.2 The PMF of a Function of a Discrete Random Variable
- 113 L11.3 A Linear Function of a Continuous Random Variable
- 114 L11.4 A Linear Function of a Normal Random Variable
- 115 L11.5 The PDF of a General Function
- 116 L11.6 The Monotonic Case
- 117 L11.7 The Intuition for the Monotonic Case
- 118 L11.8 A Nonmonotonic Example
- 119 L11.9 The PDF of a Function of Multiple Random Variables
- 120 S11.1 Simulation
- 121 L12.1 Lecture Overview
- 122 L12.2 The Sum of Independent Discrete Random Variables
- 123 L12.3 The Sum of Independent Continuous Random Variables
- 124 L12.4 The Sum of Independent Normal Random Variables
- 125 L12.5 Covariance
- 126 L12.6 Covariance Properties
- 127 L12.7 The Variance of the Sum of Random Variables
- 128 L12.8 The Correlation Coefficient
- 129 L12.9 Proof of Key Properties of the Correlation Coefficient
- 130 L12.10 Interpreting the Correlation Coefficient
- 131 L12.11 Correlations Matter
- 132 L13.1 Lecture Overview
- 133 L13.2 Conditional Expectation as a Random Variable
- 134 L13.3 The Law of Iterated Expectations
- 135 L13.4 Stick-Breaking Revisited
- 136 L13.5 Forecast Revisions
- 137 L13.6 The Conditional Variance
- 138 L13.7 Derivation of the Law of Total Variance
- 139 L13.8 A Simple Example
- 140 L13.9 Section Means and Variances
- 141 L13.10 Mean of the Sum of a Random Number of Random Variables
- 142 L13.11 Variance of the Sum of a Random Number of Random Variables
- 143 S13.1 Conditional Expectation Properties
- 144 L14.1 Lecture Overview
- 145 L14.2 Overview of Some Application Domains
- 146 L14.3 Types of Inference Problems
- 147 L14.4 The Bayesian Inference Framework
- 148 L14.5 Discrete Parameter, Discrete Observation
- 149 L14.6 Discrete Parameter, Continuous Observation
- 150 L14.7 Continuous Parameter, Continuous Observation
- 151 L14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution
- 152 L14.9 Inferring the Unknown Bias of a Coin - Point Estimates
- 153 L14.10 Summary
- 154 S14.1 The Beta Formula
- 155 L15.1 Lecture Overview
- 156 L15.2 Recognizing Normal PDFs
- 157 L15.3 Estimating a Normal Random Variable in the Presence of Additive Noise
- 158 L15.4 The Case of Multiple Observations
- 159 L15.5 The Mean Squared Error
- 160 L15.6 Multiple Parameters; Trajectory Estimation
- 161 L15.7 Linear Normal Models
- 162 L15.8 Trajectory Estimation Illustration
- 163 L16.1 Lecture Overview
- 164 L16.2 LMS Estimation in the Absence of Observations
- 165 L16.3 LMS Estimation of One Random Variable Based on Another
- 166 L16.4 LMS Performance Evaluation
- 167 L16.5 Example: The LMS Estimate
- 168 L16.6 Example Continued: LMS Performance Evaluation
- 169 L16.7 LMS Estimation with Multiple Observations or Unknowns
- 170 L16.8 Properties of the LMS Estimation Error
- 171 L17.1 Lecture Overview
- 172 L17.2 LLMS Formulation
- 173 L17.3 Solution to the LLMS Problem
- 174 L17.4 Remarks on the LLMS Solution and on the Error Variance
- 175 L17.5 LLMS Example
- 176 L17.6 LLMS for Inferring the Parameter of a Coin
- 177 L17.7 LLMS with Multiple Observations
- 178 L17.8 The Simplest LLMS Example with Multiple Observations
- 179 L17.9 The Representation of the Data Matters in LLMS
- 180 L18.1 Lecture Overview
- 181 L18.2 The Markov Inequality
- 182 L18.3 The Chebyshev Inequality
- 183 L18.4 The Weak Law of Large Numbers
- 184 L18.5 Polling
- 185 L18.6 Convergence in Probability
- 186 L18.7 Convergence in Probability Examples
- 187 L18.8 Related Topics
- 188 S18.1 Convergence in Probability of the Sum of Two Random Variables
- 189 S18.2 Jensen's Inequality
- 190 S18.3 Hoeffding's Inequality
- 191 L19.1 Lecture Overview
- 192 L19.2 The Central Limit Theorem
- 193 L19.3 Discussion of the CLT
- 194 L19.4 Illustration of the CLT
- 195 L19.5 CLT Examples
- 196 L19.6 Normal Approximation to the Binomial
- 197 L19.7 Polling Revisited
- 198 L20.1 Lecture Overview
- 199 L20.2 Overview of the Classical Statistical Framework
- 200 L20.3 The Sample Mean and Some Terminology
- 201 L20.4 On the Mean Squared Error of an Estimator
- 202 L20.5 Confidence Intervals
- 203 L20.6 Confidence Intervals for the Estimation of the Mean
- 204 L20.7 Confidence Intervals for the Mean, When the Variance is Unknown
- 205 L20.8 Other Natural Estimators
- 206 L20.9 Maximum Likelihood Estimation
- 207 L20.10 Maximum Likelihood Estimation Examples
- 208 L21.1 Lecture Overview
- 209 L21.2 The Bernoulli Process
- 210 L21.3 Stochastic Processes
- 211 L21.4 Review of Known Properties of the Bernoulli Process
- 212 L21.5 The Fresh Start Property
- 213 L21.6 Example: The Distribution of a Busy Period
- 214 L21.7 The Time of the K-th Arrival
- 215 L21.8 Merging of Bernoulli Processes
- 216 L21.9 Splitting a Bernoulli Process
- 217 L21.10 The Poisson Approximation to the Binomial
- 218 L22.1 Lecture Overview
- 219 L22.2 Definition of the Poisson Process
- 220 L22.3 Applications of the Poisson Process
- 221 L22.4 The Poisson PMF for the Number of Arrivals
- 222 L22.5 The Mean and Variance of the Number of Arrivals
- 223 L22.6 A Simple Example
- 224 L22.7 Time of the K-th Arrival
- 225 L22.8 The Fresh Start Property and Its Implications
- 226 L22.9 Summary of Results
- 227 L22.10 An Example
- 228 L23.1 Lecture Overview
- 229 L23.2 The Sum of Independent Poisson Random Variables
- 230 L23.3 Merging Independent Poisson Processes
- 231 L23.4 Where is an Arrival of the Merged Process Coming From?
- 232 L23.5 The Time Until the First (or last) Lightbulb Burns Out
- 233 L23.6 Splitting a Poisson Process
- 234 L23.7 Random Incidence in the Poisson Process
- 235 L23.8 Random Incidence in a Non-Poisson Process
- 236 L23.9 Different Sampling Methods can Give Different Results
- 237 S23.1 Poisson Versus Normal Approximations to the Binomial
- 238 S23.2 Poisson Arrivals During an Exponential Interval
- 239 L24.1 Lecture Overview
- 240 L24.2 Introduction to Markov Processes
- 241 L24.3 Checkout Counter Example
- 242 L24.4 Discrete-Time Finite-State Markov Chains
- 243 L24.5 N-Step Transition Probabilities
- 244 L24.6 A Numerical Example - Part I
- 245 L24.7 Generic Convergence Questions
- 246 L24.8 Recurrent and Transient States
- 247 L25.1 Brief Introduction (RES.6-012 Introduction to Probability)
- 248 L25.2 Lecture Overview
- 249 L25.3 Markov Chain Review
- 250 L25.4 The Probability of a Path
- 251 L25.5 Recurrent and Transient States: Review
- 252 L25.6 Periodic States
- 253 L25.7 Steady-State Probabilities and Convergence
- 254 L25.8 A Numerical Example - Part II
- 255 L25.9 Visit Frequency Interpretation of Steady-State Probabilities
- 256 L25.10 Birth-Death Processes - Part I
- 257 L25.11 Birth-Death Processes - Part II
- 258 L26.1 Brief Introduction (RES.6-012 Introduction to Probability)
- 259 L26.2 Lecture Overview
- 260 L26.3 Review of Steady-State Behavior
- 261 L26.4 A Numerical Example - Part III
- 262 L26.5 Design of a Phone System
- 263 L26.6 Absorption Probabilities
- 264 L26.7 Expected Time to Absorption
- 265 L26.8 Mean First Passage Time
- 266 L26.9 Gambler's Ruin
