This section contains lecture videos and slides. http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures 2013-01-12T04:34:05+05:00 MIT OpenCourseWare http://ocw.mit.edu en-US Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> Probability, as it appears in the real world, is related to axiomatic mathematical models. Events, independence, and random variables are reviewed, stressing both the axioms and intuition.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Axioms of probability, probability measure, events, experiments, likelihood, models, random variables, distribution function<br><br>Thumbnail - <a href= http://img.youtube.com/vi/7CYXy9J4Aao/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec01_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-1-introduction-probability/id533123931?i=117501186>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/7CYXy9J4Aao>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-1-introduction-and-probability-review Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Axioms of probability probability measure events experiments likelihood models random variables distribution function MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> The review of probability is continued with expectation, multiple random variables, and conditioning. We then move on to develop the weak law of large numbers (WLLN) and the Bernoulli process.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: expectation, indicator random variable, multiple random variables, independent identically distributed rv (IID), law of large numbers (LLN), central limit theorem (CLT), binomial distribution, Bernoulli process<br><br>Thumbnail - <a href= http://img.youtube.com/vi/d4xfax4_Iww/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec02_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-2-more-review-bernoulli/id533123931?i=116392848>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/d4xfax4_Iww>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-2-more-review-the-bernoulli-process Gallager, Robert 2011-11-03T15:28:27+05:00 en-US expectation indicator random variable multiple random variables independent identically distributed rv (IID) law of large numbers (LLN) central limit theorem (CLT) binomial distribution Bernoulli process MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture begins with the use of the WLLN in probabilistic modeling. Next the central limit theorem, the strong law of large numbers (SLLN), and convergence are discussed.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Markov, Chebychev, Chernoff, WLLN, CLT, convergence<br><br>Thumbnail - <a href= http://img.youtube.com/vi/k0UZNZwPO8Q/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec03_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-3-law-large-numbers/id533123931?i=116392856>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/k0UZNZwPO8Q>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-3-law-of-large-numbers-convergence Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Markov Chebychev Chernoff WLLN CLT convergence MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture begins with a description of arrival processes, and continues on to describe the Poisson process from three different viewpoints.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Poisson process, memoryless, exponential, Erlang, stationary and independent increment<br><br>Thumbnail - <a href= http://img.youtube.com/vi/qxaBDDib9_A/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec04_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-4-poisson-perfect/id533123931?i=116392854>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/qxaBDDib9_A>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-4-poisson-the-perfect-arrival-process Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Poisson process memoryless exponential Erlang stationary and independent increment MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description: </strong>In this lecture, many problem solving techniques are developed using, first, combining and splitting of various Poisson processes, and, second, conditioning on the number of arrivals in an interval.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Poisson process, combining Poisson, splitting Poisson, non-homogeneous Poisson processes<br><br>Thumbnail - <a href= http://img.youtube.com/vi/ct0QGoi3n4Q/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec05_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-5-poisson-combining/id533123931?i=117501187>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ct0QGoi3n4Q>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-5-poisson-combining-and-splitting Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Poisson process combining Poisson splitting Poisson non-homogeneous Poisson processes MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture treats joint conditional densities for Poisson processes and then defines finite-state Markov chains. Recurrent and transient states, periodic states, and ergodic chains are discussed.</p> <p>(Courtesy of Mina Karzand. Used with permission.)</p> <p><strong>Instructor:</strong> Mina Karzand</p>Keywords: Erlang, finite-state Markov chains, periodic states, periodic classes, ergodic Markov chains<br><br>Thumbnail - <a href= http://img.youtube.com/vi/cE6OD7DkCSU/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec06_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-6-from-poisson-to/id533123931?i=116392851>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/cE6OD7DkCSU>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-6-from-poisson-to-markov Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Erlang finite-state Markov chains periodic states periodic classes ergodic Markov chains MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> The transition matrix approach to finite-state Markov chains is developed in this lecture. The powers of the transition matrix are analyzed to understand steady-state behavior.</p> <p>(Courtesy of Shan-Yuan Ho. Used with permission.)</p> <p><strong>Instructor:</strong> Shan-Yuan Ho</p>Keywords: probability transition matrix, convergence, ergodic Markov chains, ergodic unichains, finite-state Markov chains<br><br>Thumbnail - <a href= http://img.youtube.com/vi/0aqgeLTNfQ0/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec07_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-7-finite-state-markov/id533123931?i=117127007>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/0aqgeLTNfQ0>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-7-finite-state-markov-chains-the-matrix-approach Gallager, Robert 2011-11-03T15:28:27+05:00 en-US probability transition matrix convergence ergodic Markov chains ergodic unichains finite-state Markov chains MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture covers eigenvalues and eigenvectors of the transition matrix and the steady-state vector of Markov chains. It also includes an analysis of a 2-state Markov chain and a discussion of the Jordan form.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: finite state Markov chains, transient states, recurrent state, ergodic unichain, steady state<br><br>Thumbnail - <a href= http://img.youtube.com/vi/GCFd0VVnWTw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec08_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-8-markov-eigenvalues/id533123931?i=116392849>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GCFd0VVnWTw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-8-markov-eigenvalues-and-eigenvectors Gallager, Robert 2011-11-03T15:28:27+05:00 en-US finite state Markov chains transient states recurrent state ergodic unichain steady state MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture covers rewards for Markov chains, expected first passage time, and aggregate rewards with a final reward. The professor then moves on to discuss dynamic programming and the dynamic programming algorithm.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: eigenvalues, eigenvalues dynamic programming, aggregate rewards, expected first passage, dynamic programming<br><br>Thumbnail - <a href= http://img.youtube.com/vi/mNGVkKeMUtc/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec09_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-9-markov-rewards-dynamic/id533123931?i=116392846>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/mNGVkKeMUtc>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-9-markov-rewards-and-dynamic-programming Gallager, Robert 2011-11-03T15:28:27+05:00 en-US eigenvalues eigenvalues dynamic programming aggregate rewards expected first passage dynamic programming MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> Renewal processes are introduced and their importance in analyzing other processes is explained. Proofs about convergence with probability 1 (WP1) and the SLLN are given.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: renewal process, G/G/m, SLLN, 4th moment, WP1<br><br>Thumbnail - <a href= http://img.youtube.com/vi/uHMVJJHsym4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec10_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-10-renewals-strong/id533123931?i=116392855>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/uHMVJJHsym4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-10-renewals-and-the-strong-law-of-large-numbers Gallager, Robert 2011-11-03T15:28:27+05:00 en-US renewal process G/G/m SLLN 4th moment WP1 MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture begins with the SLLN and the central limit theorem for renewal processes. This is followed by the time-average behavior of reward functions such as residual life.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: SLLN, WP1, CLT, sample paths, renewal processes, reward functions, time average residual life<br><br>Thumbnail - <a href= http://img.youtube.com/vi/QWHtRR1jMEQ/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec11_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-11-renewals-strong/id533123931?i=116392847>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/QWHtRR1jMEQ>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-11-renewals-strong-law-and-rewards Gallager, Robert 2011-11-03T15:28:27+05:00 en-US SLLN WP1 CLT sample paths renewal processes reward functions time average residual life MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> In this lecture, we learn about time-averages for renewal rewards, stopping trials for stochastic processes, and Wald's equality.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: SLLN, residual life, age time-averages, stopping trials, Wald's equality<br><br>Thumbnail - <a href= http://img.youtube.com/vi/8KQR4NAl3Iw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec12_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-12-renewal-rewards/id533123931?i=116392850>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/8KQR4NAl3Iw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-12-renewal-rewards-stopping-trials-and-walds-inequality Gallager, Robert 2011-11-03T15:28:27+05:00 en-US SLLN residual life age time-averages stopping trials Wald's equality MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture covers a variety of topics, including elementary renewal theorem, generalized stopping trials, the G/G/1 queue, Little's theorem, ensemble averages and more.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Wald, renewal theorem, G/G/1, expected wait time, queue, M/G/1, Little's theorem, ensemble averages<br><br>Thumbnail - <a href= http://img.youtube.com/vi/pOhZUJ5BQXk/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec13_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-13-little-m-g-1-ensemble/id533123931?i=116392843>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/pOhZUJ5BQXk>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-13-little-m-g-1-ensemble-averages Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Wald renewal theorem G/G/1 expected wait time queue M/G/1 Little's theorem ensemble averages MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture reviews the previous 13 lectures in preparation for the upcoming quiz.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: sample space, random variables, distribution function, convergence, WLLN, processes, CLT, arrival, Poisson, renewal, finite-state Markov chains, ergodic unichain, residual life, stopping time, Wald, Little's Theorem<br><br>Thumbnail - <a href= http://img.youtube.com/vi/pY9ol9So2Yw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec14_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-14-review/id533123931?i=116392845>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/pY9ol9So2Yw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-14-review Gallager, Robert 2011-11-03T15:28:27+05:00 en-US sample space random variables distribution function convergence WLLN processes CLT arrival Poisson renewal finite-state Markov chains ergodic unichain residual life stopping time Wald Little's Theorem MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> In this lecture, we continue our discussion of renewals and cover topics such as Markov chains and renewal processes, expected number of renewals, elementary renewal and Blackwell theorems, and delayed renewal processes.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: sample path average, Wald, Little, renewal process, Blackwell, delayed renewal process<br><br>Thumbnail - <a href= http://img.youtube.com/vi/fY7NgCWCWoQ/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec15_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-15-the-last-renewal/id533123931?i=116393420>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/fY7NgCWCWoQ>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-15-the-last-renewal Gallager, Robert 2011-11-03T15:28:27+05:00 en-US sample path average Wald Little renewal process Blackwell delayed renewal process MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> After reviewing the three major renewal theorems, we introduce Markov chains with countable state spaces. The matrix approach for finite-state chains is replaced by renewals based on first-passage times.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: renewal theorems, strong law, Blackwell, countable-state Markov chain, birth-death chains<br><br>Thumbnail - <a href= http://img.youtube.com/vi/k2PjTm1JyuI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec16_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-16-renewals-countable/id533123931?i=116393418>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/k2PjTm1JyuI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-16-renewals-and-countable-state-markov Gallager, Robert 2011-11-03T15:28:27+05:00 en-US renewal theorems strong law Blackwell countable-state Markov chain birth-death chains MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture begins with a discussion of convergence WP1 related to a quiz problem. Then positive and null recurrence, steady state, birth-death chains, and reversibility are covered.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: strong law proof, irreducible Markov chain, steady-state, positive-recurrent, null-recurrent, birth-death, reversibility<br><br>Thumbnail - <a href= http://img.youtube.com/vi/s98jdWi2kEs/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec17_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-17-countable-state/id533123931?i=116393422>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/s98jdWi2kEs>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-17-countable-state-markov-chains Gallager, Robert 2011-11-03T15:28:27+05:00 en-US strong law proof irreducible Markov chain steady-state positive-recurrent null-recurrent birth-death reversibility MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> In this lecture, the professor covers sample-time M/M/1 queue, Burke&rsquo;s theorem, branching processes, and Markov processes with countable state spaces.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: reversibility, M/M/1, branching, Burke's theorem, Markov process<br><br>Thumbnail - <a href= http://img.youtube.com/vi/_IDgYAGKyuo/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec18_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-18-countable-state/id533123931?i=116393426>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/_IDgYAGKyuo>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-18-countable-state-markov-chains-and-processes Gallager, Robert 2011-11-03T15:28:27+05:00 en-US reversibility M/M/1 branching Burke's theorem Markov process MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> Markov processes with countable state-spaces are developed in terms of the embedded Markov chain. The steady-state process probabilities and the steady-state transition probabilities are treated.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Markov processes, steady-state, irreducible, embedded Markov chain<br><br>Thumbnail - <a href= http://img.youtube.com/vi/ImKFBTqLqdE/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec19_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-19-countable-state/id533123931?i=116393423>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/ImKFBTqLqdE>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-19-countable-state-markov-processes Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Markov processes steady-state irreducible embedded Markov chain MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> After reviewing steady-state, this lecture discusses reversibility for Markov processes and for tandem M/M/1 queues. Random walks and their applications are then introduced.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: steady-state, Markov Process, reversibility, random walks, queuing delay, G/G/1 queue, detection, decision, hypothesis testing<br><br>Thumbnail - <a href= http://img.youtube.com/vi/hzJpaNcAoko/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec20_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-20-markov-processes/id533123931?i=116393417>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/hzJpaNcAoko>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-20-markov-processes-and-random-walks Gallager, Robert 2011-11-03T15:28:27+05:00 en-US steady-state Markov Process reversibility random walks queuing delay G/G/1 queue detection decision hypothesis testing MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> Sequential hypothesis testing is viewed as a random walk example. Threshold hypothesis tests are distinguished from random walk thresholds. Random walk threshold probabilities are analyzed by Chernoff bounds.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: random walks, detection, hypothesis testing, threshold tests, error curve, likelihoods, MAP, Chernoff<br><br>Thumbnail - <a href= http://img.youtube.com/vi/goT94BheP3E/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec21_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-21-hypothesis-testing/id533123931?i=116393419>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/goT94BheP3E>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-21-hypothesis-testing-and-random-walks Gallager, Robert 2011-11-03T15:28:27+05:00 en-US random walks detection hypothesis testing threshold tests error curve likelihoods MAP Chernoff MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture covers topics including the Kingman bound for G/G/1, large deviations for hypothesis tests, sequential detection, and tilted random variables and proof of Wald's identity.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Wald's identity, Chernoff bound, Kingman bound, G/G/1, large deviations, hypothesis tests, sequential detection, tilted random variables<br><br>Thumbnail - <a href= http://img.youtube.com/vi/K-iHODiS0-8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec22_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-22-random-walks-thresholds/id533123931?i=116393421>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/K-iHODiS0-8>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-22-random-walks-and-thresholds Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Wald's identity Chernoff bound Kingman bound G/G/1 large deviations hypothesis tests sequential detection tilted random variables MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> After reviewing Wald's identity, we introduce martingales and show they include many processes already studied. Next, submartingales, supermartingales, and stopped (simple, sub, super) martingales are introduced.</p> <p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Wald, sequential tests, log likelihood ratio, martingale, super-martingale, sub-martingale<br><br>Thumbnail - <a href= http://img.youtube.com/vi/GwVjWQykCDw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec23_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-23-martingales-plain/id533123931?i=117501188>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GwVjWQykCDw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-23-martingales-plain-sub-and-super Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Wald sequential tests log likelihood ratio martingale super-martingale sub-martingale MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> This lecture continues our conversation on Martingales and covers stopped martingales, Kolmogorov submartingale inequality, martingale convergence theorem, and more.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: Jensens' inequality, martingales, Kolmogorov submartingale, martingale convergence theorem<br><br>Thumbnail - <a href= http://img.youtube.com/vi/TOvSJkC1nRI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec24_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-24-martingales-stopping/id533123931?i=116393424>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/TOvSJkC1nRI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-24-martingales-stopping-and-converging Gallager, Robert 2011-11-03T15:28:27+05:00 en-US Jensens' inequality martingales Kolmogorov submartingale martingale convergence theorem MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Description:</strong> In this lecture, we put together many of the topics covered throughout the term: martingales; Markov chains; countable state Markov processes; reversibility for Markov processes; random walks; and Wald's identity for two thresholds.</p><p><strong>Instructor:</strong> Prof. Robert Gallager</p>Keywords: martingales, Kolmogorov, Markov chains, Markov processes, reversibility, random walks, Wald's identity, two thresholds<br><br>Thumbnail - <a href= http://img.youtube.com/vi/mq3nFovdG3o/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.262S11/MIT6_262S11_lec25_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-25-putting-it-all/id533123931?i=116393425>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/mq3nFovdG3o>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/video-lectures/lecture-25-putting-it-all-together Gallager, Robert 2011-11-03T15:28:27+05:00 en-US martingales Kolmogorov Markov chains Markov processes reversibility random walks Wald's identity two thresholds MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm