Theory of probability distributions | Statistical laws | Algebra of random variables

Law of total expectation

The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same probability space, then i.e., the expected value of the conditional expected value of given is the same as the expected value of . One special case states that if is a finite or countable partition of the sample space, then Note: The conditional expected value E(X | Z) is a random variable whose value depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E(X | Z = z) = g(z) then the random variable E(X | Z) is g(Z). Similar comments apply to the conditional covariance. (Wikipedia).

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4.5.5 Total Expectation: Video

MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015

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(PP 4.2) Expectation for random variables with densities

(0:00) Definition of expectation for r.v.s. with densities. (2:30) E(X) for a uniform random variable. (5:05) Well-defined expectation. (7:15) E(X) may exist and be infinite. (8:00) E(X) might fail to exist. A playlist of the Probability Primer series is available here: http://www.youtub

From playlist Probability Theory

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(PP 4.1) Expectation for discrete random variables

(0:00) Definition of expectation for discrete r.v.s. (4:17) Well-defined expectation. (8:15) E(X) may exist and be infinite. (10:58) E(X) might fail to exist. A playlist of the Probability Primer series is available here: http://www.youtube.com/view_play_list?p=17567A1A3F5DB5E4

From playlist Probability Theory

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Expectation Values in Quantum Mechanics

Expectation values in quantum mechanics are an important tool, which help us to mathematically describe measurements of quantum systems. You can think of expectation values as the average of all possible outcomes of a measurement, weighted by their respective probabilities. Contents: 00:

From playlist Quantum Mechanics, Quantum Field Theory

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(PP 4.4) Properties of expectation

(0:00) Properties of expectation. (6:17) Expectation rule. A playlist of the Probability Primer series is available here: http://www.youtube.com/view_play_list?p=17567A1A3F5DB5E4

From playlist Probability Theory

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(PP 4.3) Expectation rule

(0:00) Function of a random variable is a random variable. (1:43) Expectation rule. A playlist of the Probability Primer series is available here: http://www.youtube.com/view_play_list?p=17567A1A3F5DB5E4

From playlist Probability Theory

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Thm 1.10 - Probabilistic Version - part 06 - "Second Term"

Here we apply Jensen's inquality.

From playlist Theorem 1.10

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Derivation.3.Variance as an Expectation

This video is brought to you by the Quantitative Analysis Institute at Wellesley College. The material is best viewed as part of the online resources that organize the content and include questions for checking understanding: https://www.wellesley.edu/qai/onlineresources

From playlist Optional - Derivations

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L13.7 Derivation of the Law of Total Variance

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

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

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L13.9 Section Means and Variances

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

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

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Lec 23 | MIT 5.60 Thermodynamics & Kinetics, Spring 2008

Lecture 23: Colligative properties. View the complete course at: http://ocw.mit.edu/5-60S08 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 5.60 Thermodynamics & Kinetics, Spring 2008

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A Coin with Random Bias

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 View the complete course: http://ocw.mit.edu/6-041SCF13 Instructor: Jimmy Li License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013

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Transport in Perturbed Integrable Anharmonic Chains by Stefano Lepri

PROGRAM CLASSICAL AND QUANTUM TRANSPORT PROCESSES: CURRENT STATE AND FUTURE DIRECTIONS (ONLINE) ORGANIZERS: Alberto Imparato (University of Aarhus, Denmark), Anupam Kundu (ICTS-TIFR, India), Carlos Mejia-Monasterio (Technical University of Madrid, Spain) and Lamberto Rondoni (Polytechni

From playlist Classical and Quantum Transport Processes : Current State and Future Directions (ONLINE)2022

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Lec 20 | MIT 5.60 Thermodynamics & Kinetics, Spring 2008

Lecture 20: Phase equilibria - two components. View the complete course at: http://ocw.mit.edu/5-60S08 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 5.60 Thermodynamics & Kinetics, Spring 2008

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L13.10 Mean of the Sum of a Random Number of Random Variables

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

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

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Jeffrey Galkowski: Quantum Sabine law for resonances in transmission problems

Abstract: We prove a quantum Sabine law for the location of resonances in transmission problems. In this talk, our main applications are to scattering by strictly convex, smooth, transparent obstacles and highly frequency dependent delta potentials. In each case, we give a sharp characteri

From playlist Partial Differential Equations

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(PP 6.1) Multivariate Gaussian - definition

Introduction to the multivariate Gaussian (or multivariate Normal) distribution.

From playlist Probability Theory

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L13.11 Variance of the Sum of a Random Number of Random Variables

MIT RES.6-012 Introduction to Probability, Spring 2018 View the complete course: https://ocw.mit.edu/RES-6-012S18 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu

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

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Law of total variance | Random variable | Sample space | Law of total probability | Countable set | Expected value | Conditional convergence | Law of total cumulance | Dominated convergence theorem | Probability theory | Partition of a set | Pointwise convergence | Indicator function | Conditional expectation | Law of total covariance | Fundamental theorem of poker | Probability space | Radon–Nikodym theorem