--- title: Random Variables description: Random variables are rules that associate numerical values with the outcomes of random processes. Learners will understand the distinction between discrete and continuous variables, compute expected values, and use cumulative distribution functions to analyze probabilistic events. category: mathematics subcategory: probability-theory difficulty: beginner, intermediate, advanced url: /subject/random-variables --- # Random Variables Random variables are rules that associate numerical values with the outcomes of random processes. Learners will understand the distinction between discrete and continuous variables, compute expected values, and use cumulative distribution functions to analyze probabilistic events. ## Available Resources 1 Videos • 1 Books • 2 Courses • 6 Websites ## Websites ### 1. Khan Academy Probability Video lessons and practice exercises on random variables, expected value, binomial and geometric distributions, and the normal distribution. After working through it you can compute probabilities and expectations for standard discrete and continuous distributions. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/statistics-probability/random-variables-stats-library **Tags:** probability, random-variables, probability-distributions, statistics ### 2. Seeing Theory An interactive, D3.js-based walkthrough of probability and statistics in six chapters, from basic probability and distributions through frequentist inference, Bayesian inference and regression analysis. Parameters are manipulable, so sampling behaviour and fit are visible directly. **Difficulty:** Beginner | **Price:** Free **Link:** https://seeing-theory.brown.edu/basic-probability/index.html **Tags:** probability, statistics, data-visualization, regression, bayesian-inference ### 3. Introduction to Probability, Statistics, and Random Processes — Chapter 3: Discrete Random Variables (continues into Chapter 4: Continuous and Mixed Random Variables) **Author:** Hossein Pishro-Nik Open-access, peer-reviewed online textbook by a UMass Amherst professor. Chapters 3 and 4 define random variables as functions on a sample space, then work through PMFs, CDFs, expectation, variance, PDFs and mixed variables with many solved problems. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.probabilitycourse.com/chapter3/3_1_1_random_variables.php ### 4. Seeing Theory — Probability Distributions **Author:** Daniel Kunin Interactive Brown University visualization where you define your own discrete random variable over a uniform sample space and watch the induced distribution form, then compare discrete and continuous families with adjustable parameters, means, and variances. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://seeing-theory.brown.edu/probability-distributions/index.html **Tags:** random-variables, probability-distributions, interactive-visualization, expectation, variance ### 5. 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 ### 6. 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 ## Videos ### 1. Binomial distributions | Probabilities of probabilities, part 1 Animated walkthrough of the binomial distribution, using a product-review example to show how observed successes and failures translate into a probability density over an unknown underlying success rate. Explains where the binomial coefficient comes from. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=8idr1WZ1A7Q **Tags:** probability, binomial-distribution, bayesian-inference, visual-math ## 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. MIT RES.6-012 Introduction to Probability — Part I: The Fundamentals (Discrete and Continuous Random Variables) **Author:** John Tsitsiklis, Patrick Jaillet MIT OpenCourseWare video course with short lecture segments, solved problems and exercises. Lectures 5-13 cover PMFs, PDFs, CDFs, expectation, variance, conditioning, independence and derived distributions, so the learner can compute and reason with discrete and continuous random variables. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/pages/part-i-the-fundamentals/ ## Books ### 1. Introduction to Probability, Second Edition **Author:** Joseph K. Blitzstein, Jessica Hwang The Stat 110 textbook, free to read online from the authors. Treats random variables as functions on sample spaces, develops PMFs, CDFs, expectation and variance through story proofs, and includes hundreds of solved exercises. Chapter 3 (Random Variables and Their Distributions) and Chapter 4 (Expectation) are the clearest treatment in print of the sample-space-to-R framing, PMF/CDF relationships, and expectation via LOTUS. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1138369918?tag=edmonddante07-20 **Tags:** probability, random-variables, expectation, pmf-and-cdf, story-proofs, 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/random-variables