Statistics 110 (Probability), which has been taught at Harvard University by Joe Blitzstein (Professor of the Practice, Harvard Statistics Department) each year since 2006. Lecture videos, review materials, and over 250 practice problems with detailed solutions are provided. This course is an introduction to probability as a language and set of tools for understanding statistics, science, risk, and randomness. The ideas and methods are useful in statistics, science, engineering, economics, finance, and everyday life. Topics include the following. Basics: sample spaces and events, conditioning, Bayes' Theorem. Random variables and their distributions: distributions, moment generating functions, expectation, variance, covariance, correlation, conditional expectation. Univariate distributions: Normal, t, Binomial, Negative Binomial, Poisson, Beta, Gamma. Multivariate distributions: joint, conditional, and marginal distributions, independence, transformations, Multinomial, Multivariate Normal. Limit theorems: law of large numbers, central limit theorem. Markov chains: transition probabilities, stationary distributions, reversibility, convergence.
Sample spaces, naive definition of probability, counting, sampling.
Bose-Einstein, story proofs, Vandermonde identity, axioms of probability.
Birthday problems, properties of probability, Inclusion-exclusion, matching problem.
Law of total probability, conditional probability examples, conditional independence.
Law of total probability, conditional probability examples, conditional independence.
Gambler's ruin, first step analysis, random variables, Bernoulli, Binomial.
Random variables, CDFs, PMFs, discrete vs. continuous, Hypergeometric.
Independence, Geometric, expected values, indicator r.v.s, linearity, symmetry, fundamental bridge.
Linearity, Putnam problem, Negative Binomial, St. Petersburg paradox.
Sympathetic magic, Poisson distribution, Poisson approximation.
Discrete vs. continuous distributions, PDFs, variance, standard deviation, Uniform universality.
Moment generating functions(MGFs), hybrid Bayes' rule, Laplace's rule of sucession.
MGFs to get moments of Expo and Normal, sums of Poissons, joint distributions.
Joint, conditional, and marginal distributions, 2-D LOTUS, expected distance between Uniforms, chicken-egg.
Expected distance between Normals, Multinomial, Cauchy.
Covariance, correlation, variance of a sum, variance of Hypergeometric.
Transformations, LogNormal, convolutions, proving existence.
Beta distribution, Bayes' billards, finance preview and examples.
Gamma distribution, Poisson processes.
Beta-Gamma(bank-post office), order statistics, conditional expectation, two envelope paradox.
Two envelope paradox(cont.), conditional expectation(cont.), waiting for HT vs. waiting for HH.
Conditional expectation(cont.), taking out what's known, Adam's law, Eve's law, projection picture.
Sum of random numbers of random variables, inequalities(Cauchy-Schwarz, Jensen, Markov, Chebyshev).
Law of large numbers, central limit theorem.
Chi-Square, Student-t, Multivariate Normal.
Markov chains(cont.), irreducibility, recurrence, transience, reversibility, random walk on an undirected network.
Markov chains(cont.), Google PageRank as a Markov chain.
A look ahead, final review, other statistics courses, regression example, sampling from a finite population example.