Probability

Author

Chao Ma

Published

April 8, 2026

Notes on probability foundations, from sample spaces and axioms to counting, continuous models, and random phenomena.

MIT 6.041 Probability: Iterated Expectations Conditional expectation as a random variable, the law of iterated expectations, total variance, section-mean variance decomposition, and random sums.

MIT 6.041 Probability: Derived Distributions and Covariance Ratio distributions, infinite expectations, monotone transformations, convolution, sums of independent normals, covariance, variance of sums, and correlation.

MIT 6.041 Probability: Continuous Bayes’ Rule and Derived Distributions Continuous Bayes’ rule, mixed discrete-continuous inference, derived distributions, CDF methods, and linear transformations of PDFs.

MIT 6.041 Probability: Multiple Continuous Random Variables Joint PDFs, marginalization, independence, conditional densities, Buffon’s needle, and a two-step stick-breaking example.

MIT 6.041 Probability: Continuous Random Variables Continuous random variables through PDFs, CDFs, means, variances, mixed distributions, Gaussian PDFs, linear transformations, and standardization.

MIT 6.041 Probability: Discrete Random Variables III Joint PMFs for multiple random variables, multiplication rules, expectations of functions, variance of sums, binomial mean and variance, and the hat-check problem.

MIT 6.041 Probability: Discrete Random Variables II Conditional PMFs and conditional expectation, the memoryless property of the geometric distribution, total expectation, and joint PMFs.

MIT 6.041 Probability: Discrete Random Variables I Random variables as functions, probability mass functions, geometric and binomial examples, expectation as weighted average, functions of random variables, and variance.

MIT 6.041 Probability: Counting Uniform counting probabilities, permutations, subsets, binomial coefficients, binomial probabilities, and partition-based counting arguments.

MIT 6.041 Probability: Independence Independence as information-free probability, disjoint versus independent events, hidden-variable conditional independence, and why pairwise independence does not imply mutual independence.

MIT 6.041 Probability: Conditioning and Bayes’ Rule Conditional probability as belief revision, multiplication and total-probability rules, and Bayes’ rule through die and radar examples from MIT 6.041.

MIT 6.041 Probability: Probability Models and Axioms Sample spaces, discrete and continuous models, probability axioms, uniform laws, and counting examples from the opening lecture of MIT 6.041.

--- title: "Probability" author: "Chao Ma" date: "2026-04-08" --- Notes on probability foundations, from sample spaces and axioms to counting, continuous models, and random phenomena. --- ::: {.content-grid} ::: {.content-card} **[MIT 6.041 Probability: Iterated Expectations](iterated-expectations.qmd)** Conditional expectation as a random variable, the law of iterated expectations, total variance, section-mean variance decomposition, and random sums. ::: ::: {.content-card} **[MIT 6.041 Probability: Derived Distributions and Covariance](derived-distributions-covariance.qmd)** Ratio distributions, infinite expectations, monotone transformations, convolution, sums of independent normals, covariance, variance of sums, and correlation. ::: ::: {.content-card} **[MIT 6.041 Probability: Continuous Bayes' Rule and Derived Distributions](continuous-bayes-rule-derived-distributions.qmd)** Continuous Bayes' rule, mixed discrete-continuous inference, derived distributions, CDF methods, and linear transformations of PDFs. ::: ::: {.content-card} **[MIT 6.041 Probability: Multiple Continuous Random Variables](multiple-continuous-random-variables.qmd)** Joint PDFs, marginalization, independence, conditional densities, Buffon's needle, and a two-step stick-breaking example. ::: ::: {.content-card} **[MIT 6.041 Probability: Continuous Random Variables](continuous-random-variables.qmd)** Continuous random variables through PDFs, CDFs, means, variances, mixed distributions, Gaussian PDFs, linear transformations, and standardization. ::: ::: {.content-card} **[MIT 6.041 Probability: Discrete Random Variables III](discrete-random-variables-3.qmd)** Joint PMFs for multiple random variables, multiplication rules, expectations of functions, variance of sums, binomial mean and variance, and the hat-check problem. ::: ::: {.content-card} **[MIT 6.041 Probability: Discrete Random Variables II](discrete-random-variables-2.qmd)** Conditional PMFs and conditional expectation, the memoryless property of the geometric distribution, total expectation, and joint PMFs. ::: ::: {.content-card} **[MIT 6.041 Probability: Discrete Random Variables I](discrete-random-variables-1.qmd)** Random variables as functions, probability mass functions, geometric and binomial examples, expectation as weighted average, functions of random variables, and variance. ::: ::: {.content-card} **[MIT 6.041 Probability: Counting](counting.qmd)** Uniform counting probabilities, permutations, subsets, binomial coefficients, binomial probabilities, and partition-based counting arguments. ::: ::: {.content-card} **[MIT 6.041 Probability: Independence](independence.qmd)** Independence as information-free probability, disjoint versus independent events, hidden-variable conditional independence, and why pairwise independence does not imply mutual independence. ::: ::: {.content-card} **[MIT 6.041 Probability: Conditioning and Bayes' Rule](conditioning-bayes-rule.qmd)** Conditional probability as belief revision, multiplication and total-probability rules, and Bayes' rule through die and radar examples from MIT 6.041. ::: ::: {.content-card} **[MIT 6.041 Probability: Probability Models and Axioms](probability-models-axioms.qmd)** Sample spaces, discrete and continuous models, probability axioms, uniform laws, and counting examples from the opening lecture of MIT 6.041. ::: :::