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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

Includes audio/video content: AV lectures. Welcome to 6.041/6.431, a subject on the modeling and analysis of random phenomena and processes, including the basics of statistical inference. Nowadays, there is broad consensus that the ability to think probabilistically is a fundamental component of scientific literacy. For example: The concept of statistical significance (to be touched upon at the end of this course) is considered by the Financial Times as one of "The Ten Things Everyone Should Know About Science". A recent Scientific American article argues that statistical literacy is crucial in making health-related decisions. Finally, an article in the New York Times identifies statistical data analysis as an upcoming profession, valuable everywhere, from Google and Includes audio/video content: AV lectures. Welcome to 6.041/6.431, a subject on the modeling and analysis of random phenomena and processes, including the basics of statistical inference. Nowadays, there is broad consensus that the ability to think probabilistically is a fundamental component of scientific literacy. For example: The concept of statistical significance (to be touched upon at the end of this course) is considered by the Financial Times as one of "The Ten Things Everyone Should Know About Science". A recent Scientific American article argues that statistical literacy is crucial in making health-related decisions. Finally, an article in the New York Times identifies statistical data analysis as an upcoming profession, valuable everywhere, from Google and

Subjects

probability | probability | probability models | probability models | bayes rule | bayes rule | discrete random variables | discrete random variables | continuous random variables | continuous random variables | bernoulli process | bernoulli process | poisson process | poisson process | markov chains | markov chains | central limit theorem | central limit theorem | statistical inference | statistical inference

License

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6.041SC Probabilistic Systems Analysis and Applied Probability (MIT) 6.041SC Probabilistic Systems Analysis and Applied Probability (MIT)

Description

Includes audio/video content: AV lectures. This course introduces students to the modeling, quantification, and analysis of uncertainty.  The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management. Includes audio/video content: AV lectures. This course introduces students to the modeling, quantification, and analysis of uncertainty.  The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management.

Subjects

probability | probability | probability models | probability models | bayes rule | bayes rule | discrete random variables | discrete random variables | continuous random variables | continuous random variables | bernoulli process | bernoulli process | poisson process | poisson process | markov chains | markov chains | central limit theorem | central limit theorem | statistical inference | statistical inference

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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18.440 Probability and Random Variables (MIT) 18.440 Probability and Random Variables (MIT)

Description

This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem. This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

Subjects

Probability spaces | Probability spaces | random variables | random variables | distribution functions | distribution functions | Binomial | Binomial | geometric | geometric | hypergeometric | hypergeometric | Poisson distributions | Poisson distributions | Uniform | Uniform | exponential | exponential | normal | normal | gamma and beta distributions | gamma and beta distributions | Conditional probability | Conditional probability | Bayes theorem | Bayes theorem | joint distributions | joint distributions | Chebyshev inequality | Chebyshev inequality | law of large numbers | law of large numbers | and central limit theorem. | and central limit theorem.

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 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

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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference. This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference.

Subjects

probabilistic systems | probabilistic systems | probabilistic systems analysis | probabilistic systems analysis | applied probability | applied probability | uncertainty | uncertainty | uncertainty modeling | uncertainty modeling | uncertainty quantification | uncertainty quantification | analysis of uncertainty | analysis of uncertainty | uncertainty analysis | uncertainty analysis | sample space | sample space | random variables | random variables | transform techniques | transform techniques | simple random processes | simple random processes | probability distribution | probability distribution | Markov process | Markov process | limit theorem | limit theorem | statistical inference | statistical inference

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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ESD.70J Engineering Economy Module (MIT) ESD.70J Engineering Economy Module (MIT)

Description

This intensive micro-subject provides the necessary skills in Microsoft® Excel spreadsheet modeling for ESD.71 – Engineering Systems Analysis for Design. Its purpose is to bring entering students up to speed on some of the advanced techniques that we routinely use in analysis. It is motivated by our experience that many students only have an introductory knowledge of Excel, and thus waste a lot of time thrashing about unproductively. Many people think they know Excel, but overlook many efficient tools – such as Data Table and Goal Seek. It is also useful for a variety of other subjects.NoteThis MIT OpenCourseWare site is based on the materials from Professor de Neufville's ESD.70J Web site. This intensive micro-subject provides the necessary skills in Microsoft® Excel spreadsheet modeling for ESD.71 – Engineering Systems Analysis for Design. Its purpose is to bring entering students up to speed on some of the advanced techniques that we routinely use in analysis. It is motivated by our experience that many students only have an introductory knowledge of Excel, and thus waste a lot of time thrashing about unproductively. Many people think they know Excel, but overlook many efficient tools – such as Data Table and Goal Seek. It is also useful for a variety of other subjects.NoteThis MIT OpenCourseWare site is based on the materials from Professor de Neufville's ESD.70J Web site.

Subjects

excel | excel | spreadsheet | spreadsheet | modeling | modeling | dynamic modeling | dynamic modeling | analysis | analysis | data table | data table | goal seek | goal seek | sensitivity analysis | sensitivity analysis | simulation | simulation | random number generator | random number generator | counting | counting | modeling uncertainties | modeling uncertainties | random variables | random variables | statistical package | statistical package | flexibility | flexibility | contingency rules | contingency rules | excel solver | excel solver | solver | solver

License

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18.440 Probability and Random Variables (MIT) 18.440 Probability and Random Variables (MIT)

Description

This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem. This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

Subjects

Probability spaces | Probability spaces | random variables | random variables | distribution functions | distribution functions | Binomial | Binomial | geometric | geometric | hypergeometric | hypergeometric | Poisson distributions | Poisson distributions | Uniform | Uniform | exponential | exponential | normal | normal | gamma and beta distributions | gamma and beta distributions | Conditional probability | Conditional probability | Bayes theorem | Bayes theorem | joint distributions | joint distributions | Chebyshev inequality | Chebyshev inequality | law of large numbers | law of large numbers | central limit theorem | central limit theorem

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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18.175 Theory of Probability (MIT) 18.175 Theory of Probability (MIT)

Description

This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory. This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory.

Subjects

Earth | Earth | Solar System | Solar System | Geophysics | Geophysics | Gravitational Field | Gravitational Field | Magnetic Field | Magnetic Field | Seismology | Seismology | Geodynamics | Geodynamics | Laws of large numbers | Laws of large numbers | central limit theorems for sums of independent random variables | central limit theorems for sums of independent random variables | conditioning and martingales | conditioning and martingales | Brownian motion and elements of diffusion theory | Brownian motion and elements of diffusion theory | functional limit theorems | functional limit theorems

License

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1.010 Uncertainty in Engineering (MIT) 1.010 Uncertainty in Engineering (MIT)

Description

This undergraduate class serves as an introduction to probability and statistics, with emphasis on engineering applications. The first segment discusses events and their probability, Bayes' Theorem, discrete and continuous random variables and vectors, univariate and multivariate distributions, Bernoulli trials and Poisson point processes, and full-distribution uncertainty propagation and conditional analysis. The second segment deals with second-moment representation of uncertainty and second-moment uncertainty propagation and conditional analysis. The final segment covers random sampling, point and interval estimation, hypothesis testing, and linear regression. Many of the concepts covered in class are illustrated with real-world examples from various areas of engineering. This undergraduate class serves as an introduction to probability and statistics, with emphasis on engineering applications. The first segment discusses events and their probability, Bayes' Theorem, discrete and continuous random variables and vectors, univariate and multivariate distributions, Bernoulli trials and Poisson point processes, and full-distribution uncertainty propagation and conditional analysis. The second segment deals with second-moment representation of uncertainty and second-moment uncertainty propagation and conditional analysis. The final segment covers random sampling, point and interval estimation, hypothesis testing, and linear regression. Many of the concepts covered in class are illustrated with real-world examples from various areas of engineering.

Subjects

statistics | statistics | decision analysis | decision analysis | random variables and vectors | random variables and vectors | uncertainty propagation | uncertainty propagation | conditional distributions | conditional distributions | second-moment analysis | second-moment analysis | system reliability | system reliability | Bayesian analysis and risk-based decision | Bayesian analysis and risk-based decision | estimation of distribution parameters | estimation of distribution parameters | hypothesis testing | hypothesis testing | simple and multiple linear regressions | simple and multiple linear regressions | Poisson and Markov processes | Poisson and Markov processes

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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18.175 Theory of Probability (MIT) 18.175 Theory of Probability (MIT)

Description

This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory. This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory.

Subjects

Earth | Earth | Solar System | Solar System | Geophysics | Geophysics | Gravitational Field | Gravitational Field | Magnetic Field | Magnetic Field | Seismology | Seismology | Geodynamics | Geodynamics | Laws of large numbers | Laws of large numbers | central limit theorems for sums of independent random variables | central limit theorems for sums of independent random variables | conditioning and martingales | conditioning and martingales | Brownian motion and elements of diffusion theory | Brownian motion and elements of diffusion theory | functional limit theorems | functional limit theorems

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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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference. This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference.

Subjects

probabilistic systems | probabilistic systems | probabilistic systems analysis | probabilistic systems analysis | applied probability | applied probability | uncertainty | uncertainty | uncertainty modeling | uncertainty modeling | uncertainty quantification | uncertainty quantification | analysis of uncertainty | analysis of uncertainty | uncertainty analysis | uncertainty analysis | sample space | sample space | random variables | random variables | transform techniques | transform techniques | simple random processes | simple random processes | probability distribution | probability distribution | Markov process | Markov process | limit theorem | limit theorem | statistical inference | statistical inference

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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1.010 Uncertainty in Engineering (MIT) 1.010 Uncertainty in Engineering (MIT)

Description

This course gives an introduction to probability and statistics, with emphasis on engineering applications. Course topics include events and their probability, the total probability and Bayes' theorems, discrete and continuous random variables and vectors, uncertainty propagation and conditional analysis. Second-moment representation of uncertainty, random sampling, estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation), and simple and multiple linear regression. Concepts illustrated with examples from various areas of engineering and everyday life. This course gives an introduction to probability and statistics, with emphasis on engineering applications. Course topics include events and their probability, the total probability and Bayes' theorems, discrete and continuous random variables and vectors, uncertainty propagation and conditional analysis. Second-moment representation of uncertainty, random sampling, estimation of distribution parameters (method of moments, maximum likelihood, Bayesian estimation), and simple and multiple linear regression. Concepts illustrated with examples from various areas of engineering and everyday life.

Subjects

fundamentals of probability | fundamentals of probability | random processes | random processes | statistics | statistics | decision analysis | decision analysis | random variables and vectors | random variables and vectors | uncertainty propagation | uncertainty propagation | conditional distributions | conditional distributions | second-moment analysis | second-moment analysis | system reliability | system reliability | Bayes theorem | Bayes theorem | total probability theorem | total probability theorem | Bayesian analysis and risk-based decision | Bayesian analysis and risk-based decision | estimation of distribution parameters | estimation of distribution parameters | hypothesis testing | hypothesis testing | simple and multiple linear regressions | simple and multiple linear regressions | Poisson and Markov processes | Poisson and Markov processes

License

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1.203J Logistical and Transportation Planning Methods (MIT) 1.203J Logistical and Transportation Planning Methods (MIT)

Description

The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics. The class will cover quantitative techniques of Operations Research with emphasis on applications in transportation systems analysis (urban, air, ocean, highway, pick-up and delivery systems) and in the planning and design of logistically oriented urban service systems (e.g., fire and police departments, emergency medical services, emergency repair services). It presents a unified study of functions of random variables, geometrical probability, multi-server queueing theory, spatial location theory, network analysis and graph theory, and relevant methods of simulation. There will be discussion focused on the difficulty of implementation, among other topics.

Subjects

1.203 | 1.203 | 6.281 | 6.281 | 15.073 | 15.073 | 16.76 | 16.76 | ESD.216 | ESD.216 | logistics | logistics | transportation | transportation | hypercube models | hypercube models | barrier example | barrier example | operations research | operations research | spatial queues | spatial queues | queueing models | queueing models | network models | network models | TSP | TSP | heuristics | heuristics | geometrical probabilities | geometrical probabilities | Markov | Markov | quantitative techniques | quantitative techniques | transportation systems analysis | transportation systems analysis | urban service systems | urban service systems | emergency services | emergency services | random variables | random variables | multi-server queueing theory | multi-server queueing theory | spatial location theory | spatial location theory | network analysis | network analysis | graph theory | graph theory | simulation | simulation | urban OR | urban OR

License

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1.151 Probability and Statistics in Engineering (MIT) 1.151 Probability and Statistics in Engineering (MIT)

Description

This class covers quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, random processes, statistics, and decision analysis are covered, along with random variables and vectors, uncertainty propagation, conditional distributions, and second-moment analysis. System reliability is introduced. Other topics covered include Bayesian analysis and risk-based decision, estimation of distribution parameters, hypothesis testing, simple and multiple linear regressions, and Poisson and Markov processes. There is an emphasis placed on real-world applications to engineering problems. This class covers quantitative analysis of uncertainty and risk for engineering applications. Fundamentals of probability, random processes, statistics, and decision analysis are covered, along with random variables and vectors, uncertainty propagation, conditional distributions, and second-moment analysis. System reliability is introduced. Other topics covered include Bayesian analysis and risk-based decision, estimation of distribution parameters, hypothesis testing, simple and multiple linear regressions, and Poisson and Markov processes. There is an emphasis placed on real-world applications to engineering problems.

Subjects

fundamentals of probability | fundamentals of probability | random processes | random processes | statistics | statistics | decision analysis | decision analysis | random variables and vectors | random variables and vectors | uncertainty propagation | uncertainty propagation | conditional distributions | conditional distributions | second-moment analysis | second-moment analysis | system reliability | system reliability | Bayesian analysis and risk-based decision | Bayesian analysis and risk-based decision | estimation of distribution parameters | estimation of distribution parameters | hypothesis testing | hypothesis testing | simple and multiple linear regressions | simple and multiple linear regressions | Poisson and Markov processes | Poisson and Markov processes

License

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1.017 Computing and Data Analysis for Environmental Applications (MIT) 1.017 Computing and Data Analysis for Environmental Applications (MIT)

Description

This subject is a computer-oriented introduction to probability and data analysis. It is designed to give students the knowledge and practical experience they need to interpret lab and field data. Basic probability concepts are introduced at the outset because they provide a systematic way to describe uncertainty. They form the basis for the analysis of quantitative data in science and engineering. The MATLAB® programming language is used to perform virtual experiments and to analyze real-world data sets, many downloaded from the web. Programming applications include display and assessment of data sets, investigation of hypotheses, and identification of possible casual relationships between variables. This is the first semester that two courses, Computing and Data Analysis for Environm This subject is a computer-oriented introduction to probability and data analysis. It is designed to give students the knowledge and practical experience they need to interpret lab and field data. Basic probability concepts are introduced at the outset because they provide a systematic way to describe uncertainty. They form the basis for the analysis of quantitative data in science and engineering. The MATLAB® programming language is used to perform virtual experiments and to analyze real-world data sets, many downloaded from the web. Programming applications include display and assessment of data sets, investigation of hypotheses, and identification of possible casual relationships between variables. This is the first semester that two courses, Computing and Data Analysis for Environm

Subjects

probability | probability | statistics | statistics | events | events | random variables | random variables | univariate distributions | univariate distributions | multivariate distributions | multivariate distributions | uncertainty propagation | uncertainty propagation | Bernoulli trials | Bernoulli trials | Poisson processed | Poisson processed | conditional probability | conditional probability | Bayes rule | Bayes rule | random sampling | random sampling | point estimation | point estimation | interval estimation | interval estimation | hypothesis testing | hypothesis testing | analysis of variance | analysis of variance | linear regression | linear regression | computational analysis | computational analysis | data analysis | data analysis | environmental engineering | environmental engineering | applications | applications | MATLAB | MATLAB | numerical modeling | numerical modeling | probabilistic concepts | probabilistic concepts | statistical methods | statistical methods | field data | field data | laboratory data | laboratory data | numerical techniques | numerical techniques | Monte Carlo simulation | Monte Carlo simulation | variability | variability | sampling | sampling | data sets | data sets | computer | computer | uncertainty | uncertainty | interpretation | interpretation | quantitative data | quantitative data

License

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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

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6.041 Probabilistic Systems Analysis and Applied Probability (MIT) 6.041 Probabilistic Systems Analysis and Applied Probability (MIT)

Description

This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference. This course is offered both to undergraduates (6.041) and graduates (6.431), but the assignments differ. 6.041/6.431 introduces students to the modeling, quantification, and analysis of uncertainty. Topics covered include: formulation and solution in sample space, random variables, transform techniques, simple random processes and their probability distributions, Markov processes, limit theorems, and elements of statistical inference.

Subjects

probabilistic systems | probabilistic systems | probabilistic systems analysis | probabilistic systems analysis | applied probability | applied probability | uncertainty | uncertainty | uncertainty modeling | uncertainty modeling | uncertainty quantification | uncertainty quantification | analysis of uncertainty | analysis of uncertainty | uncertainty analysis | uncertainty analysis | sample space | sample space | random variables | random variables | transform techniques | transform techniques | simple random processes | simple random processes | probability distribution | probability distribution | Markov process | Markov process | limit theorem | limit theorem | statistical inference | statistical inference

License

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9.07 Statistical Methods in Brain and Cognitive Science (MIT) 9.07 Statistical Methods in Brain and Cognitive Science (MIT)

Description

This course emphasizes statistics as a powerful tool for studying complex issues in behavioral and biological sciences, and explores the limitations of statistics as a method of inquiry. The course covers descriptive statistics, probability and random variables, inferential statistics, and basic issues in experimental design. Techniques introduced include confidence intervals, t-tests, F-tests, regression, and analysis of variance. Assignments include a project in data analysis. This course emphasizes statistics as a powerful tool for studying complex issues in behavioral and biological sciences, and explores the limitations of statistics as a method of inquiry. The course covers descriptive statistics, probability and random variables, inferential statistics, and basic issues in experimental design. Techniques introduced include confidence intervals, t-tests, F-tests, regression, and analysis of variance. Assignments include a project in data analysis.

Subjects

statistics | statistics | standard deviation | standard deviation | ANOVA | ANOVA | variance | variance | chi squared | chi squared | mean | mean | median | median | spread | spread | graphs | graphs | histograms | histograms | binomial distribution | binomial distribution | random variables | random variables | sampling | sampling | experimental design | experimental design | probability | probability | confidence intervals | confidence intervals | error bars | error bars | best fit | best fit | hypothesis testing | hypothesis testing | linear regression | linear regression | regression | regression | correlation | correlation | categorical data | categorical data

License

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16.322 Stochastic Estimation and Control (MIT) 16.322 Stochastic Estimation and Control (MIT)

Description

The major themes of this course are estimation and control of dynamic systems. Preliminary topics begin with reviews of probability and random variables. Next, classical and state-space descriptions of random processes and their propagation through linear systems are introduced, followed by frequency domain design of filters and compensators. From there, the Kalman filter is employed to estimate the states of dynamic systems. Concluding topics include conditions for stability of the filter equations. The major themes of this course are estimation and control of dynamic systems. Preliminary topics begin with reviews of probability and random variables. Next, classical and state-space descriptions of random processes and their propagation through linear systems are introduced, followed by frequency domain design of filters and compensators. From there, the Kalman filter is employed to estimate the states of dynamic systems. Concluding topics include conditions for stability of the filter equations.

Subjects

probability | probability | stochastic estimation | stochastic estimation | estimation | estimation | random variables | random variables | random processes | random processes | state space | state space | Wiener filter | Wiener filter | control system design | control system design | Kalman filter | Kalman filter

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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ESD.70J Engineering Economy Module (MIT) ESD.70J Engineering Economy Module (MIT)

Description

This intensive micro-subject provides the necessary skills in Microsoft® Excel spreadsheet modeling for ESD.71 Engineering Systems Analysis for Design. Its purpose is to bring entering students up to speed on some of the advanced techniques that we routinely use in analysis. It is motivated by our experience that many students only have an introductory knowledge of Excel, and thus waste a lot of time thrashing about unproductively. Many people think they know Excel, but overlook many efficient tools, such as Data Table and Goal Seek. It is also useful for a variety of other subjects. This intensive micro-subject provides the necessary skills in Microsoft® Excel spreadsheet modeling for ESD.71 Engineering Systems Analysis for Design. Its purpose is to bring entering students up to speed on some of the advanced techniques that we routinely use in analysis. It is motivated by our experience that many students only have an introductory knowledge of Excel, and thus waste a lot of time thrashing about unproductively. Many people think they know Excel, but overlook many efficient tools, such as Data Table and Goal Seek. It is also useful for a variety of other subjects.

Subjects

ESD.70 | ESD.70 | 1.145 | 1.145 | excel | excel | spreadsheet | spreadsheet | modeling | modeling | dynamic modeling | dynamic modeling | analysis | analysis | data table | data table | goal seek | goal seek | sensitivity analysis | sensitivity analysis | simulation | simulation | random number generator | random number generator | counting | counting | modeling uncertainties | modeling uncertainties | random variables | random variables | statistical package | statistical package | flexibility | flexibility | contingency rules | contingency rules | excel solver | excel solver | solver | solver

License

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8.044 Statistical Physics I (MIT) 8.044 Statistical Physics I (MIT)

Description

This course offers an introduction to probability, statistical mechanics, and thermodynamics. Numerous examples are used to illustrate a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices.This course is an elective subject in MIT’s undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges. This course offers an introduction to probability, statistical mechanics, and thermodynamics. Numerous examples are used to illustrate a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices.This course is an elective subject in MIT’s undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges.

Subjects

probability | probability | statistical mechanics | statistical mechanics | thermodynamics | thermodynamics | random variables | random variables | joint and conditional probability densities | joint and conditional probability densities | functions of a random variable | functions of a random variable | macroscopic variables | macroscopic variables | thermodynamic equilibrium | thermodynamic equilibrium | fundamental assumption of statistical mechanics | fundamental assumption of statistical mechanics | microcanonical and canonical ensembles | microcanonical and canonical ensembles | First | second | and third laws of thermodynamics | First | second | and third laws of thermodynamics | magnetism | magnetism | polyatomic gases | polyatomic gases | thermal radiation | thermal radiation | electrons in solids | electrons in solids | noise in electronic devices | noise in electronic devices

License

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18.175 Theory of Probability (MIT) 18.175 Theory of Probability (MIT)

Description

This course covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. This course covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.

Subjects

Laws of large numbers | Laws of large numbers | central limit theorems | central limit theorems | independent random variables | independent random variables | conditioning | conditioning | martingales | martingales | Brownian motion | Brownian motion | elements of diffusion theory | elements of diffusion theory | functional limit theorems | functional limit theorems

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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8.044 Statistical Physics I (MIT) 8.044 Statistical Physics I (MIT)

Description

This course offers an introduction to probability, statistical mechanics, and thermodynamics. Numerous examples are used to illustrate a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices. This course offers an introduction to probability, statistical mechanics, and thermodynamics. Numerous examples are used to illustrate a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices.

Subjects

probability | probability | statistical mechanics | statistical mechanics | thermodynamics | thermodynamics | random variables | random variables | joint and conditional probability densities | joint and conditional probability densities | functions of a random variable | functions of a random variable | macroscopic variables | macroscopic variables | thermodynamic equilibrium | thermodynamic equilibrium | fundamental assumption of statistical mechanics | fundamental assumption of statistical mechanics | microcanonical and canonical ensembles | microcanonical and canonical ensembles | First | First | second | second | and third laws of thermodynamics | and third laws of thermodynamics | magnetism | magnetism | polyatomic gases | polyatomic gases | hermal radiation | hermal radiation | thermal radiation | thermal radiation | electrons in solids | electrons in solids | and noise in electronic devices | and noise in electronic devices | First | second | and third laws of thermodynamics | First | second | and third laws of thermodynamics

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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8.044 Statistical Physics I (MIT) 8.044 Statistical Physics I (MIT)

Description

Introduction to probability, statistical mechanics, and thermodynamics. Random variables, joint and conditional probability densities, and functions of a random variable. Concepts of macroscopic variables and thermodynamic equilibrium, fundamental assumption of statistical mechanics, microcanonical and canonical ensembles. First, second, and third laws of thermodynamics. Numerous examples illustrating a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices. Concurrent enrollment in 8.04, Quantum Physics I, is recommended. Introduction to probability, statistical mechanics, and thermodynamics. Random variables, joint and conditional probability densities, and functions of a random variable. Concepts of macroscopic variables and thermodynamic equilibrium, fundamental assumption of statistical mechanics, microcanonical and canonical ensembles. First, second, and third laws of thermodynamics. Numerous examples illustrating a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices. Concurrent enrollment in 8.04, Quantum Physics I, is recommended.

Subjects

probability | probability | statistical mechanics | statistical mechanics | thermodynamics | thermodynamics | random variables | random variables | joint and conditional probability densities | joint and conditional probability densities | functions of a random variable | functions of a random variable | macroscopic variables | macroscopic variables | thermodynamic equilibrium | thermodynamic equilibrium | fundamental assumption of statistical mechanics | fundamental assumption of statistical mechanics | microcanonical and canonical ensembles | microcanonical and canonical ensembles | First | First | second | second | and third laws of thermodynamics | and third laws of thermodynamics | magnetism | magnetism | polyatomic gases | polyatomic gases | hermal radiation | hermal radiation | thermal radiation | thermal radiation | electrons in solids | electrons in solids | and noise in electronic devices | and noise in electronic devices | First | second | and third laws of thermodynamics | First | second | and third laws of thermodynamics

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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18.440 Probability and Random Variables (MIT) 18.440 Probability and Random Variables (MIT)

Description

This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem. This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

Subjects

Probability spaces | Probability spaces | random variables | random variables | distribution functions | distribution functions | Binomial | Binomial | geometric | geometric | hypergeometric | hypergeometric | Poisson distributions | Poisson distributions | Uniform | Uniform | exponential | exponential | normal | normal | gamma and beta distributions | gamma and beta distributions | Conditional probability | Conditional probability | Bayes theorem | Bayes theorem | joint distributions | joint distributions | Chebyshev inequality | Chebyshev inequality | law of large numbers | law of large numbers | And central limit theorem | And central limit theorem

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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