--- title: Expected Value description: Expected value is the long-run average value of a random variable over many repetitions of an experiment. Learners will understand how to calculate and interpret expected outcomes for discrete and continuous distributions. category: mathematics subcategory: probability-theory difficulty: beginner, intermediate, advanced url: /subject/expected-value --- # Expected Value Expected value is the long-run average value of a random variable over many repetitions of an experiment. Learners will understand how to calculate and interpret expected outcomes for discrete and continuous distributions. ## Available Resources 1 Videos • 1 Books • 3 Courses • 4 Websites ## Websites ### 1. Khan Academy Expected Value Free lesson from Khan Academy's statistics and probability course combining short videos and practice exercises on the expected value of discrete random variables. Learners compute a distribution's mean from its probability table and interpret expected value in simple games and decisions. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library/expected-value-lib **Tags:** expected-value, probability, discrete-random-variables, statistics ### 2. Expected Value — Statlect **Author:** Marco Taboga Reference lecture defining expected value three ways: the discrete sum, the continuous integral, and the general Riemann-Stieltjes and Lebesgue formulations. Covers linearity, scaling, existence conditions for non-integrable variables, and solved exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.statlect.com/fundamentals-of-probability/expected-value **Tags:** expected-value, probability-theory, lebesgue-integral, random-variables, measure-theory ### 3. probabilitycourse.com ProbabilityCourse.com is an online learning resource that provides a structured probability course with clear explanations, worked examples, and practice problems covering topics from basics to advanced theory. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://probabilitycourse.com **Tags:** websites, technology-computer-science, data-science--ai ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Probabilistic Systems Analysis and Applied Probability (MIT 6.041SC) **Author:** John Tsitsiklis Modeling and analysis of uncertainty: probability models, discrete and continuous random variables, Bayesian inference, limit theorems and random processes. Designed for independent study, with lecture videos, slides, recitation and tutorial problems, problem sets and exams with solutions, and TA problem-solving videos. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/ **Tags:** probability, random-variables, bayesian-inference, limit-theorems, random-processes, statistical-inference ### 2. An Intuitive Introduction to Probability **Author:** Karl Schmedders University of Zurich course taught by Karl Schmedders that builds probability from everyday examples: basic rules, conditional probability, applications, discrete random variables, and the normal distribution. Learners finish able to compute and interpret probabilities in practical decisions. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/introductiontoprobability **Tags:** probability, conditional-probability, discrete-random-variables, normal-distribution ### 3. Introduction to Probability **Author:** Joe Blitzstein HarvardX's self-paced version of Harvard's Stat 110, taught by Joe Blitzstein. Covers counting and story proofs, conditional probability and Bayes' rule, discrete and continuous random variables, joint distributions, the law of large numbers, the central limit theorem and Markov chains. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/probability/harvard-university-introduction-to-probability **Tags:** probability, random-variables, bayes-rule, probability-distributions, markov-chains, central-limit-theorem ## Videos ### 1. Lecture 5: Discrete Random Variables; Probability Mass Functions; Expectations (MIT 6.041SC) **Author:** John Tsitsiklis Defines discrete random variables and their probability mass functions, including binomial and geometric examples, then introduces expected value and the expected value rule. The recorded lecture prepares learners to compute expectations for common discrete distributions and functions of random variables. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/resources/lecture-5-video-2/ **Tags:** expected-value, random-variables, probability-mass-functions, discrete-distributions, probability ## Books ### 1. Grinstead and Snell's Introduction to Probability (Chapter 6: Expected Value and Variance) **Author:** Charles M. Grinstead, J. Laurie Snell Free 500-page probability textbook released under the GNU Free Documentation License. Chapter 6, Expected Value and Variance, covers expectation for discrete and continuous random variables, linearity, and historical problems including the St. Petersburg paradox, with exercises. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://math.dartmouth.edu/~prob/prob/prob.pdf **Tags:** probability, expected-value, variance, random-variables, open-textbook ## Youtubes ### 1. Statistics 110 Lecture 9: Expectation, Indicator Random Variables, Linearity **Author:** Joseph K. Blitzstein Joe Blitzstein's Harvard lecture defines expectation and builds the core computational toolkit: indicator random variables, linearity of expectation, and symmetry arguments, worked through examples including the geometric and binomial distributions. The single best hour anywhere on this exact concept: it does not merely define E[X] but teaches the two techniques that make expectation usable (indicators plus linearity without independence). **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=LX2q356N2rU **Tags:** expected-value, linearity-of-expectation, indicator-random-variables, probability, harvard-stat-110 --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/expected-value