Probability
Notes on probability foundations, from sample spaces and axioms to counting, continuous models, and random phenomena.
MIT 6.041SC Lecture 23: Classical Statistical Inference I Maximum likelihood, estimator quality, and repeated-sampling confidence intervals with unknown variance.
MIT 6.041SC Lecture 22: Bayesian Statistical Inference II Posterior uncertainty, error orthogonality, optimal affine estimators, precision-weighted measurements, and feature representations.
MIT 6.041SC Lecture 21: Bayesian Statistical Inference I Prior and likelihood, posterior uncertainty, MAP decisions, and the squared-error meaning of conditional expectation.
MIT 6.041SC Lecture 20: Central Limit Theorem Standardized sums, a sharper polling estimate, the continuity correction, and normal versus Poisson limits.
MIT 6.041SC Lecture 19: Weak Law of Large Numbers Chebyshev bounds, the weak law, a conservative polling guarantee, and a rare-tail counterexample.
MIT 6.041 Probability: Markov Chains III Erlang loss capacity, absorption probabilities, and first hitting versus first return times.
MIT 6.041 Probability: Markov Chains II Steady-state convergence, balance equations, birth–death chains, and queue growth near capacity.
MIT 6.041 Probability: Markov Chains I Choosing a sufficient state, transition recursions, long-run probabilities, and recurrent versus transient states.
MIT 6.041 Probability: Poisson Process II Poisson stopping rules, competing exponential lifetimes, memorylessness, and length-biased bus intervals observed at a random time.
MIT 6.041 Probability: Poisson Process I Poisson processes, the Bernoulli-to-Poisson limit, interarrival times, Erlang waiting times, and merging independent Poisson processes.
MIT 6.041 Probability: Bernoulli Process Bernoulli processes, binomial counts, geometric first-arrival times, the time of the kth success, and splitting and merging arrival streams.
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.