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Playlist · MIT OpenCourseWare

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

by MIT OpenCourseWare · 266 videos

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1 L01.1 Lecture Overview 1:52 2018-04-24
2 L01.2 Sample Space 5:38 2018-04-24
3 L01.3 Sample Space Examples 5:03 2018-04-24
4 L01.4 Probability Axioms 8:55 2018-04-24
5 L01.5 Simple Properties of Probabilities 11:05 2018-04-24
6 L01.6 More Properties of Probabilities 8:40 2018-04-24
7 L01.7 A Discrete Example 5:13 2018-04-24
8 L01.8 A Continuous Example 5:20 2018-04-24
9 L01.9 Countable Additivity 12:10 2018-04-24
10 L01.10 Interpretations & Uses of Probabilities 3:48 2018-04-24
11 S01.0 Mathematical Background Overview 1:25 2018-04-24
12 S01.1 Sets 10:55 2018-04-24
13 S01.2 De Morgan's Laws 4:53 2018-04-24
14 S01.3 Sequences and their Limits 6:00 2018-04-24
15 S01.4 When Does a Sequence Converge 2:46 2018-04-24
16 S01.5 Infinite Series 3:11 2018-04-24
17 S01.6 The Geometric Series 4:07 2018-04-24
18 S01.7 About the Order of Summation in Series with Multiple Indices 10:05 2018-04-24
19 S01.8 Countable and Uncountable Sets 6:19 2018-04-24
20 S01.9 Proof That a Set of Real Numbers is Uncountable 4:02 2018-04-24
21 S01.10 Bonferroni's Inequality 9:28 2018-04-24
22 L02.1 Lecture Overview 2:07 2018-04-24
23 L02.2 Conditional Probabilities 9:00 2018-04-24
24 L02.3 A Die Roll Example 5:02 2018-04-24
25 L02.4 Conditional Probabilities Obey the Same Axioms 7:45 2018-04-24
26 L02.5 A Radar Example and Three Basic Tools 10:59 2018-04-24
27 L02.6 The Multiplication Rule 6:17 2018-04-24
28 L02.7 Total Probability Theorem 5:25 2018-04-24
29 L02.8 Bayes' Rule 4:28 2018-04-24
30 L03.1 Lecture Overview 1:27 2018-04-24
31 L03.2 A Coin Tossing Example 7:59 2018-04-24
32 L03.3 Independence of Two Events 6:10 2018-04-24
33 L03.4 Independence of Event Complements 2:59 2018-04-24
34 L03.5 Conditional Independence 2:46 2018-04-24
35 L03.6 Independence Versus Conditional Independence 5:30 2018-04-24
36 L03.7 Independence of a Collection of Events 6:00 2018-04-24
37 L03.8 Independence Versus Pairwise Independence 8:35 2018-04-24
38 L03.9 Reliability 7:28 2018-04-24
39 L03.10 The King's Sibling 6:54 2018-04-24
40 L04.1 Lecture Overview 2:29 2018-04-24
41 L04.2 The Counting Principle 11:12 2018-04-24
42 L04.3 Die Roll Example 4:39 2018-04-24
43 L04.4 Combinations 10:08 2018-04-24
44 L04.5 Binomial Probabilities 6:38 2018-04-24
45 L04.6 A Coin Tossing Example 11:48 2018-04-24
46 L04.7 Partitions 5:20 2018-04-24
47 L04.8 Each Person Gets An Ace 9:45 2018-04-24
48 L04.9 Multinomial Probabilities 10:36 2018-04-24
49 L05.1 Lecture Overview 1:40 2018-04-24
50 L05.2 Definition of Random Variables 9:14 2018-04-24
51 L05.3 Probability Mass Functions 10:21 2018-04-24
52 L05.4 Bernoulli & Indicator Random Variables 3:06 2018-04-24
53 L05.5 Uniform Random Variables 4:06 2018-04-24
54 L05.6 Binomial Random Variables 6:08 2018-04-24
55 L05.7 Geometric Random Variables 7:37 2018-04-24
56 L05.8 Expectation 10:38 2018-04-24
57 L05.9 Elementary Properties of Expectation 4:12 2018-04-24
58 L05.10 The Expected Value Rule 10:00 2018-04-24
59 L05.11 Linearity of Expectations 3:59 2018-04-24
60 S05.1 Supplement: Functions 8:08 2018-04-24
61 L06.1 Lecture Overview 2:02 2018-04-24
62 L06.2 Variance 10:43 2018-04-24
63 L06.3 The Variance of the Bernoulli & The Uniform 8:40 2018-04-24
64 L06.4 Conditional PMFs & Expectations Given an Event 7:31 2018-04-24
65 L06.5 Total Expectation Theorem 6:28 2018-04-24
66 L06.6 Geometric PMF Memorylessness & Expectation 10:29 2018-04-24
67 L06.7 Joint PMFs and the Expected Value Rule 10:16 2018-04-24
68 L06.8 Linearity of Expectations & The Mean of the Binomial 8:25 2018-04-24
69 L07.1 Lecture Overview 1:50 2018-04-24
70 L07.2 Conditional PMFs 10:48 2018-04-24
71 L07.3 Conditional Expectation & the Total Expectation Theorem 6:10 2018-04-24
72 L07.4 Independence of Random Variables 5:08 2018-04-24
73 L07.5 Example 4:44 2018-04-24
74 L07.6 Independence & Expectations 4:22 2018-04-24
75 L07.7 Independence, Variances & the Binomial Variance 7:09 2018-04-24
76 L07.8 The Hat Problem 16:09 2018-04-24
77 S07.1 The Inclusion-Exclusion Formula 11:13 2018-04-24
78 S07.2 The Variance of the Geometric 5:42 2018-04-24
79 S07.3 Independence of Random Variables Versus Independence of Events 6:51 2018-04-24
80 L08.1 Lecture Overview 1:13 2018-04-24
81 L08.2 Probability Density Functions 11:09 2018-04-24
82 L08.3 Uniform & Piecewise Constant PDFs 2:52 2018-04-24
83 L08.4 Means & Variances 6:57 2018-04-24
84 L08.5 Mean & Variance of the Uniform 3:56 2018-04-24
85 L08.6 Exponential Random Variables 8:09 2018-04-24
86 L08.7 Cumulative Distribution Functions 12:48 2018-04-24
87 L08.8 Normal Random Variables 9:14 2018-04-24
88 L08.9 Calculation of Normal Probabilities 10:11 2018-04-24
89 L09.1 Lecture Overview 1:33 2018-04-24
90 L09.2 Conditioning A Continuous Random Variable on an Event 9:56 2018-04-24
91 L09.3 Conditioning Example 3:08 2018-04-24
92 L09.4 Memorylessness of the Exponential PDF 8:18 2018-04-24
93 L09.5 Total Probability & Expectation Theorems 6:51 2018-04-24
94 L09.6 Mixed Random Variables 5:35 2018-04-24
95 L09.7 Joint PDFs 9:18 2018-04-24
96 L09.8 From The Joint to the Marginal 7:23 2018-04-24
97 L09.9 Continuous Analogs of Various Properties 1:40 2018-04-24
98 L09.10 Joint CDFs 4:16 2018-04-24
99 S09.1 Buffon's Needle & Monte Carlo Simulation 16:12 2018-04-24
100 L10.1 Lecture Overview 1:42 2018-04-24
101 L10.2 Conditional PDFs 6:57 2018-04-24
102 L10.3 Comments on Conditional PDFs 4:34 2018-04-24
103 L10.4 Total Probability & Total Expectation Theorems 5:17 2018-04-24
104 L10.5 Independence 3:35 2018-04-24
105 L10.6 Stick-Breaking Example 10:02 2018-04-24
106 L10.7 Independent Normals 5:36 2018-04-24
107 L10.8 Bayes Rule Variations 3:27 2018-04-24
108 L10.9 Mixed Bayes Rule 8:33 2018-04-24
109 L10.10 Detection of a Binary Signal 9:15 2018-04-24
110 L10.11 Inference of the Bias of a Coin 6:00 2018-04-24
111 L11.1 Lecture Overview 1:52 2018-04-24
112 L11.2 The PMF of a Function of a Discrete Random Variable 6:42 2018-04-24
113 L11.3 A Linear Function of a Continuous Random Variable 11:18 2018-04-24
114 L11.4 A Linear Function of a Normal Random Variable 2:45 2018-04-24
115 L11.5 The PDF of a General Function 9:47 2018-04-24
116 L11.6 The Monotonic Case 11:07 2018-04-24
117 L11.7 The Intuition for the Monotonic Case 5:28 2018-04-24
118 L11.8 A Nonmonotonic Example 7:14 2018-04-24
119 L11.9 The PDF of a Function of Multiple Random Variables 7:42 2018-04-24
120 S11.1 Simulation 12:35 2018-04-24
121 L12.1 Lecture Overview 1:29 2018-04-24
122 L12.2 The Sum of Independent Discrete Random Variables 7:52 2018-04-24
123 L12.3 The Sum of Independent Continuous Random Variables 6:45 2018-04-24
124 L12.4 The Sum of Independent Normal Random Variables 3:10 2018-04-24
125 L12.5 Covariance 5:54 2018-04-24
126 L12.6 Covariance Properties 5:48 2018-04-24
127 L12.7 The Variance of the Sum of Random Variables 5:36 2018-04-24
128 L12.8 The Correlation Coefficient 7:03 2018-04-24
129 L12.9 Proof of Key Properties of the Correlation Coefficient 3:52 2018-04-24
130 L12.10 Interpreting the Correlation Coefficient 5:50 2018-04-24
131 L12.11 Correlations Matter 6:22 2018-04-24
132 L13.1 Lecture Overview 1:47 2018-04-24
133 L13.2 Conditional Expectation as a Random Variable 4:31 2018-04-24
134 L13.3 The Law of Iterated Expectations 3:58 2018-04-24
135 L13.4 Stick-Breaking Revisited 3:53 2018-04-24
136 L13.5 Forecast Revisions 4:38 2018-04-24
137 L13.6 The Conditional Variance 5:02 2018-04-24
138 L13.7 Derivation of the Law of Total Variance 4:54 2018-04-24
139 L13.8 A Simple Example 6:29 2018-04-24
140 L13.9 Section Means and Variances 9:04 2018-04-24
141 L13.10 Mean of the Sum of a Random Number of Random Variables 6:26 2018-04-24
142 L13.11 Variance of the Sum of a Random Number of Random Variables 5:10 2018-04-24
143 S13.1 Conditional Expectation Properties 8:13 2018-04-24
144 L14.1 Lecture Overview 2:10 2018-04-24
145 L14.2 Overview of Some Application Domains 5:17 2018-04-24
146 L14.3 Types of Inference Problems 5:24 2018-04-24
147 L14.4 The Bayesian Inference Framework 9:48 2018-04-24
148 L14.5 Discrete Parameter, Discrete Observation 6:46 2018-04-24
149 L14.6 Discrete Parameter, Continuous Observation 4:35 2018-04-24
150 L14.7 Continuous Parameter, Continuous Observation 3:46 2018-04-24
151 L14.8 Inferring the Unknown Bias of a Coin and the Beta Distribution 7:35 2018-04-24
152 L14.9 Inferring the Unknown Bias of a Coin - Point Estimates 9:30 2018-04-24
153 L14.10 Summary 5:41 2018-04-24
154 S14.1 The Beta Formula 10:24 2018-04-24
155 L15.1 Lecture Overview 1:59 2018-04-24
156 L15.2 Recognizing Normal PDFs 7:15 2018-04-24
157 L15.3 Estimating a Normal Random Variable in the Presence of Additive Noise 8:18 2018-04-24
158 L15.4 The Case of Multiple Observations 13:47 2018-04-24
159 L15.5 The Mean Squared Error 13:02 2018-04-24
160 L15.6 Multiple Parameters; Trajectory Estimation 10:32 2018-04-24
161 L15.7 Linear Normal Models 5:12 2018-04-24
162 L15.8 Trajectory Estimation Illustration 10:55 2018-04-24
163 L16.1 Lecture Overview 1:13 2018-04-24
164 L16.2 LMS Estimation in the Absence of Observations 6:48 2018-04-24
165 L16.3 LMS Estimation of One Random Variable Based on Another 9:24 2018-04-24
166 L16.4 LMS Performance Evaluation 4:32 2018-04-24
167 L16.5 Example: The LMS Estimate 6:31 2018-04-24
168 L16.6 Example Continued: LMS Performance Evaluation 5:29 2018-04-24
169 L16.7 LMS Estimation with Multiple Observations or Unknowns 5:24 2018-04-24
170 L16.8 Properties of the LMS Estimation Error 5:59 2018-04-24
171 L17.1 Lecture Overview 1:41 2018-04-24
172 L17.2 LLMS Formulation 4:58 2018-04-24
173 L17.3 Solution to the LLMS Problem 5:06 2018-04-24
174 L17.4 Remarks on the LLMS Solution and on the Error Variance 8:02 2018-04-24
175 L17.5 LLMS Example 6:43 2018-04-24
176 L17.6 LLMS for Inferring the Parameter of a Coin 11:29 2018-04-24
177 L17.7 LLMS with Multiple Observations 6:54 2018-04-24
178 L17.8 The Simplest LLMS Example with Multiple Observations 5:06 2018-04-24
179 L17.9 The Representation of the Data Matters in LLMS 7:03 2018-04-24
180 L18.1 Lecture Overview 1:57 2018-04-24
181 L18.2 The Markov Inequality 10:21 2018-04-24
182 L18.3 The Chebyshev Inequality 5:57 2018-04-24
183 L18.4 The Weak Law of Large Numbers 7:31 2018-04-24
184 L18.5 Polling 8:12 2018-04-24
185 L18.6 Convergence in Probability 8:28 2018-04-24
186 L18.7 Convergence in Probability Examples 8:05 2018-04-24
187 L18.8 Related Topics 6:44 2018-04-24
188 S18.1 Convergence in Probability of the Sum of Two Random Variables 10:13 2018-04-24
189 S18.2 Jensen's Inequality 12:19 2018-04-24
190 S18.3 Hoeffding's Inequality 18:28 2018-04-24
191 L19.1 Lecture Overview 1:50 2018-04-24
192 L19.2 The Central Limit Theorem 6:58 2018-04-24
193 L19.3 Discussion of the CLT 9:00 2018-04-24
194 L19.4 Illustration of the CLT 2:54 2018-04-24
195 L19.5 CLT Examples 13:56 2018-04-24
196 L19.6 Normal Approximation to the Binomial 11:53 2018-04-24
197 L19.7 Polling Revisited 13:54 2018-04-24
198 L20.1 Lecture Overview 2:46 2018-04-24
199 L20.2 Overview of the Classical Statistical Framework 11:00 2018-04-24
200 L20.3 The Sample Mean and Some Terminology 4:58 2018-04-24
201 L20.4 On the Mean Squared Error of an Estimator 6:53 2018-04-24
202 L20.5 Confidence Intervals 5:04 2018-04-24
203 L20.6 Confidence Intervals for the Estimation of the Mean 4:27 2018-04-24
204 L20.7 Confidence Intervals for the Mean, When the Variance is Unknown 6:13 2018-04-24
205 L20.8 Other Natural Estimators 4:37 2018-04-24
206 L20.9 Maximum Likelihood Estimation 6:32 2018-04-24
207 L20.10 Maximum Likelihood Estimation Examples 10:20 2018-04-24
208 L21.1 Lecture Overview 2:01 2018-04-24
209 L21.2 The Bernoulli Process 4:21 2018-04-24
210 L21.3 Stochastic Processes 6:21 2018-04-24
211 L21.4 Review of Known Properties of the Bernoulli Process 2:20 2018-04-24
212 L21.5 The Fresh Start Property 11:26 2018-04-24
213 L21.6 Example: The Distribution of a Busy Period 4:16 2018-04-24
214 L21.7 The Time of the K-th Arrival 8:12 2018-04-24
215 L21.8 Merging of Bernoulli Processes 7:12 2018-04-24
216 L21.9 Splitting a Bernoulli Process 5:54 2018-04-24
217 L21.10 The Poisson Approximation to the Binomial 6:12 2018-04-24
218 L22.1 Lecture Overview 1:31 2018-04-24
219 L22.2 Definition of the Poisson Process 5:07 2018-04-24
220 L22.3 Applications of the Poisson Process 3:03 2018-04-24
221 L22.4 The Poisson PMF for the Number of Arrivals 8:01 2018-04-24
222 L22.5 The Mean and Variance of the Number of Arrivals 3:22 2018-04-24
223 L22.6 A Simple Example 3:07 2018-04-24
224 L22.7 Time of the K-th Arrival 10:41 2018-04-24
225 L22.8 The Fresh Start Property and Its Implications 10:33 2018-04-24
226 L22.9 Summary of Results 2:34 2018-04-24
227 L22.10 An Example 14:08 2018-04-24
228 L23.1 Lecture Overview 1:39 2018-04-24
229 L23.2 The Sum of Independent Poisson Random Variables 4:03 2018-04-24
230 L23.3 Merging Independent Poisson Processes 8:22 2018-04-24
231 L23.4 Where is an Arrival of the Merged Process Coming From? 5:00 2018-04-24
232 L23.5 The Time Until the First (or last) Lightbulb Burns Out 11:25 2018-04-24
233 L23.6 Splitting a Poisson Process 5:06 2018-04-24
234 L23.7 Random Incidence in the Poisson Process 9:09 2018-04-24
235 L23.8 Random Incidence in a Non-Poisson Process 4:36 2018-04-24
236 L23.9 Different Sampling Methods can Give Different Results 3:59 2018-04-24
237 S23.1 Poisson Versus Normal Approximations to the Binomial 8:56 2018-04-24
238 S23.2 Poisson Arrivals During an Exponential Interval 9:37 2018-04-24
239 L24.1 Lecture Overview 1:59 2018-04-24
240 L24.2 Introduction to Markov Processes 2:09 2018-04-24
241 L24.3 Checkout Counter Example 12:10 2018-04-24
242 L24.4 Discrete-Time Finite-State Markov Chains 7:53 2018-04-24
243 L24.5 N-Step Transition Probabilities 10:59 2018-04-24
244 L24.6 A Numerical Example - Part I 9:26 2018-04-24
245 L24.7 Generic Convergence Questions 5:32 2018-04-24
246 L24.8 Recurrent and Transient States 5:37 2018-04-24
247 L25.1 Brief Introduction (RES.6-012 Introduction to Probability) 1:40 2018-04-24
248 L25.2 Lecture Overview 1:05 2018-04-24
249 L25.3 Markov Chain Review 6:15 2018-04-24
250 L25.4 The Probability of a Path 6:39 2018-04-24
251 L25.5 Recurrent and Transient States: Review 3:26 2018-04-24
252 L25.6 Periodic States 6:49 2018-04-24
253 L25.7 Steady-State Probabilities and Convergence 9:13 2018-04-24
254 L25.8 A Numerical Example - Part II 3:58 2018-04-24
255 L25.9 Visit Frequency Interpretation of Steady-State Probabilities 5:19 2018-04-24
256 L25.10 Birth-Death Processes - Part I 8:56 2018-04-24
257 L25.11 Birth-Death Processes - Part II 8:57 2018-04-24
258 L26.1 Brief Introduction (RES.6-012 Introduction to Probability) 1:41 2018-04-24
259 L26.2 Lecture Overview 0:40 2018-04-24
260 L26.3 Review of Steady-State Behavior 9:12 2018-04-24
261 L26.4 A Numerical Example - Part III 10:35 2018-04-24
262 L26.5 Design of a Phone System 18:30 2018-04-24
263 L26.6 Absorption Probabilities 9:58 2018-04-24
264 L26.7 Expected Time to Absorption 11:30 2018-04-24
265 L26.8 Mean First Passage Time 8:44 2018-04-24
266 L26.9 Gambler's Ruin 11:24 2018-04-24
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