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6.436J Fundamentals of Probability (MIT) 6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in MIT course 6.431 but at a faster pace and in more depth. Topics covered include: probability spaces and measures; discrete and continuous random variables; conditioning and independence; multivariate normal distribution; abstract integration, expectation, and related convergence results; moment generating and characteristic functions; Bernoulli and Poisson processes; finite-state Markov chains; convergence notions and their relations; and limit theorems. Familiarity with elementary notions in probability and real analysis is desirable. This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in MIT course 6.431 but at a faster pace and in more depth. Topics covered include: probability spaces and measures; discrete and continuous random variables; conditioning and independence; multivariate normal distribution; abstract integration, expectation, and related convergence results; moment generating and characteristic functions; Bernoulli and Poisson processes; finite-state Markov chains; convergence notions and their relations; and limit theorems. Familiarity with elementary notions in probability and real analysis is desirable.

Subjects

Introduction to probability theory | Introduction to probability theory | Probability spaces and measures | Probability spaces and measures | Discrete and continuous random variables | Discrete and continuous random variables | Conditioning and independence | Conditioning and independence | Multivariate normal distribution | Multivariate normal distribution | Abstract integration | Abstract integration | expectation | expectation | and related convergence results | and related convergence results | Moment generating and characteristic functions | Moment generating and characteristic functions | Bernoulli and Poisson process | Bernoulli and Poisson process | Finite-state Markov chains | Finite-state Markov chains | Convergence notions and their relations | Convergence notions and their relations | Limit theorems | Limit theorems | Familiarity with elementary notions in probability and real analysis is desirable | Familiarity with elementary notions in probability and real analysis is desirable

License

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

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6.231 Dynamic Programming and Stochastic Control (MIT) 6.231 Dynamic Programming and Stochastic Control (MIT)

Description

This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed. This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed.

Subjects

dynamic programming | dynamic programming | | stochastic control | | stochastic control | | mathematics | optimization | | | mathematics | optimization | | algorithms | | algorithms | | probability | | probability | | Markov chains | | Markov chains | | optimal control | optimal control | stochastic control | stochastic control | mathematics | mathematics | optimization | optimization | algorithms | algorithms | probability | probability | Markov chains | Markov chains

License

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

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6.262 Discrete Stochastic Processes (MIT) 6.262 Discrete Stochastic Processes (MIT)

Description

Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance. Includes audio/video content: AV lectures. Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Subjects

probability | probability | Poisson processes | Poisson processes | finite-state Markov chains | finite-state Markov chains | renewal processes | renewal processes | countable-state Markov chains | countable-state Markov chains | Markov processes | Markov processes | countable state spaces | countable state spaces | random walks | random walks | large deviations | large deviations | martingales | martingales

License

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

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6.436J Fundamentals of Probability (MIT) 6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations. This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations.

Subjects

sample space | sample space | random variables | random variables | expectations | expectations | transforms | transforms | Bernoulli process | Bernoulli process | Poisson process | Poisson process | Markov chains | Markov chains | limit theorems | limit theorems | measure theory | measure theory

License

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

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6.856J Randomized Algorithms (MIT) 6.856J Randomized Algorithms (MIT)

Description

This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms. This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

Subjects

Randomized Algorithms | Randomized Algorithms | algorithms | algorithms | efficient in time and space | efficient in time and space | randomization | randomization | computational problems | computational problems | data structures | data structures | graph algorithms | graph algorithms | optimization | optimization | geometry | geometry | Markov chains | Markov chains | sampling | sampling | estimation | estimation | geometric algorithms | geometric algorithms | parallel and distributed algorithms | parallel and distributed algorithms | parallel and ditributed algorithm | parallel and ditributed algorithm | parallel and distributed algorithm | parallel and distributed algorithm | random sampling | random sampling | random selection of witnesses | random selection of witnesses | symmetry breaking | symmetry breaking | randomized computational models | randomized computational models | hash tables | hash tables | skip lists | skip lists | minimum spanning trees | minimum spanning trees | shortest paths | shortest paths | minimum cuts | minimum cuts | convex hulls | convex hulls | linear programming | linear programming | fixed dimension | fixed dimension | arbitrary dimension | arbitrary dimension | approximate counting | approximate counting | parallel algorithms | parallel algorithms | online algorithms | online algorithms | derandomization techniques | derandomization techniques | probabilistic analysis | probabilistic analysis | computational number theory | computational number theory | simplicity | simplicity | speed | speed | design | design | basic probability theory | basic probability theory | application | application | randomized complexity classes | randomized complexity classes | game-theoretic techniques | game-theoretic techniques | Chebyshev | Chebyshev | moment inequalities | moment inequalities | limited independence | limited independence | coupon collection | coupon collection | occupancy problems | occupancy problems | tail inequalities | tail inequalities | Chernoff bound | Chernoff bound | conditional expectation | conditional expectation | probabilistic method | probabilistic method | random walks | random walks | algebraic techniques | algebraic techniques | probability amplification | probability amplification | sorting | sorting | searching | searching | combinatorial optimization | combinatorial optimization | approximation | approximation | counting problems | counting problems | distributed algorithms | distributed algorithms | 6.856 | 6.856 | 18.416 | 18.416

License

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

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18.445 Introduction to Stochastic Processes (MIT) 18.445 Introduction to Stochastic Processes (MIT)

Description

This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix. This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.

Subjects

probability | probability | Stochastic Processes | Stochastic Processes | Markov chains | Markov chains | random walks | random walks | martingales | martingales | Galton-Watsom tree | Galton-Watsom tree | linear algebra | linear algebra

License

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

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2.852 Manufacturing Systems Analysis (MIT) 2.852 Manufacturing Systems Analysis (MIT)

Description

This course covers the following topics: models of manufacturing systems, including transfer lines and flexible manufacturing systems; calculation of performance measures, including throughput, in-process inventory, and meeting production commitments; real-time control of scheduling; effects of machine failure, set-ups, and other disruptions on system performance. This course covers the following topics: models of manufacturing systems, including transfer lines and flexible manufacturing systems; calculation of performance measures, including throughput, in-process inventory, and meeting production commitments; real-time control of scheduling; effects of machine failure, set-ups, and other disruptions on system performance.

Subjects

transfer lines | transfer lines | flexible manufacturing systems | flexible manufacturing systems | performance measures | performance measures | throughput | throughput | in-process inventory | in-process inventory | real-time scheduling | real-time scheduling | machine failure | machine failure | buffer design | buffer design | optimization | optimization | probability | probability | Markov chains | Markov chains | long lines | long lines | quality/quantity | quality/quantity | loops | loops | assembly/disassembly systems | assembly/disassembly systems

License

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

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6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in MIT course 6.431 but at a faster pace and in more depth. Topics covered include: probability spaces and measures; discrete and continuous random variables; conditioning and independence; multivariate normal distribution; abstract integration, expectation, and related convergence results; moment generating and characteristic functions; Bernoulli and Poisson processes; finite-state Markov chains; convergence notions and their relations; and limit theorems. Familiarity with elementary notions in probability and real analysis is desirable.

Subjects

Introduction to probability theory | Probability spaces and measures | Discrete and continuous random variables | Conditioning and independence | Multivariate normal distribution | Abstract integration | expectation | and related convergence results | Moment generating and characteristic functions | Bernoulli and Poisson process | Finite-state Markov chains | Convergence notions and their relations | Limit theorems | Familiarity with elementary notions in probability and real analysis is desirable

License

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 https://ocw.mit.edu/terms/index.htm

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6.231 Dynamic Programming and Stochastic Control (MIT)

Description

This course covers the basic models and solution techniques for problems of sequential decision making under uncertainty (stochastic control). Approximation methods for problems involving large state spaces are also presented and discussed.

Subjects

dynamic programming | | stochastic control | | mathematics | optimization | | algorithms | | probability | | Markov chains | | optimal control | stochastic control | mathematics | optimization | algorithms | probability | Markov chains

License

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 https://ocw.mit.edu/terms/index.htm

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18.445 Introduction to Stochastic Processes (MIT)

Description

This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.

Subjects

probability | Stochastic Processes | Markov chains | random walks | martingales | Galton-Watsom tree | linear algebra

License

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 https://ocw.mit.edu/terms/index.htm

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6.262 Discrete Stochastic Processes (MIT)

Description

Discrete stochastic processes are essentially probabilistic systems that evolve in time via random changes occurring at discrete fixed or random intervals. This course aims to help students acquire both the mathematical principles and the intuition necessary to create, analyze, and understand insightful models for a broad range of these processes. The range of areas for which discrete stochastic-process models are useful is constantly expanding, and includes many applications in engineering, physics, biology, operations research and finance.

Subjects

probability | Poisson processes | finite-state Markov chains | renewal processes | countable-state Markov chains | Markov processes | countable state spaces | random walks | large deviations | martingales

License

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 https://ocw.mit.edu/terms/index.htm

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6.436J Fundamentals of Probability (MIT)

Description

This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations.

Subjects

sample space | random variables | expectations | transforms | Bernoulli process | Poisson process | Markov chains | limit theorems | measure theory

License

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 https://ocw.mit.edu/terms/index.htm

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6.856J Randomized Algorithms (MIT)

Description

This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimum cuts); geometric algorithms (convex hulls, linear programming in fixed or arbitrary dimension); approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

Subjects

Randomized Algorithms | algorithms | efficient in time and space | randomization | computational problems | data structures | graph algorithms | optimization | geometry | Markov chains | sampling | estimation | geometric algorithms | parallel and distributed algorithms | parallel and ditributed algorithm | parallel and distributed algorithm | random sampling | random selection of witnesses | symmetry breaking | randomized computational models | hash tables | skip lists | minimum spanning trees | shortest paths | minimum cuts | convex hulls | linear programming | fixed dimension | arbitrary dimension | approximate counting | parallel algorithms | online algorithms | derandomization techniques | probabilistic analysis | computational number theory | simplicity | speed | design | basic probability theory | application | randomized complexity classes | game-theoretic techniques | Chebyshev | moment inequalities | limited independence | coupon collection | occupancy problems | tail inequalities | Chernoff bound | conditional expectation | probabilistic method | random walks | algebraic techniques | probability amplification | sorting | searching | combinatorial optimization | approximation | counting problems | distributed algorithms | 6.856 | 18.416

License

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 https://ocw.mit.edu/terms/index.htm

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2.852 Manufacturing Systems Analysis (MIT)

Description

This course covers the following topics: models of manufacturing systems, including transfer lines and flexible manufacturing systems; calculation of performance measures, including throughput, in-process inventory, and meeting production commitments; real-time control of scheduling; effects of machine failure, set-ups, and other disruptions on system performance.

Subjects

transfer lines | flexible manufacturing systems | performance measures | throughput | in-process inventory | real-time scheduling | machine failure | buffer design | optimization | probability | Markov chains | long lines | quality/quantity | loops | assembly/disassembly systems

License

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 https://ocw.mit.edu/terms/index.htm

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