# Statistics 110: Probability

Harvard University

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.

• ##### Probability and Counting
Joseph Blitzstein

Sample spaces, naive definition of probability, counting, sampling.

• ##### Story Proofs, Axioms of Probability
Joseph Blitzstein

Bose-Einstein, story proofs, Vandermonde identity, axioms of probability.

• ##### Birthday Problem, Properties of Probability
Joseph Blitzstein

Birthday problems, properties of probability, Inclusion-exclusion, matching problem.

• ##### Conditional Probability
Joseph Blitzstein

Law of total probability, conditional probability examples, conditional independence.

• ##### Conditioning Continued, Law of Total Probability
Joseph Blitzstein

Law of total probability, conditional probability examples, conditional independence.

• ##### Monty hall, Simpson's Paradox
Joseph Blitzstein

• ##### Gambler's Ruin and Random Variables
Joseph Blitzstein

Gambler's ruin, first step analysis, random variables, Bernoulli, Binomial.

• ##### Random Variables and Their Distributions
Joseph Blitzstein

Random variables, CDFs, PMFs, discrete vs. continuous, Hypergeometric.

• ##### Expectation, Indicator Random Variables, Linearity
Joseph Blitzstein

Independence, Geometric, expected values, indicator r.v.s, linearity, symmetry, fundamental bridge.

• ##### Expectation Continued
Joseph Blitzstein

Linearity, Putnam problem, Negative Binomial, St. Petersburg paradox.

• ##### The Poisson Distribution
Joseph Blitzstein

Sympathetic magic, Poisson distribution, Poisson approximation.

• ##### Discrete vs. Continuous, the Uniform
Joseph Blitzstein

Discrete vs. continuous distributions, PDFs, variance, standard deviation, Uniform universality.

• ##### Normal Distribution
Joseph Blitzstein

Standard Normal, Normal normalizing constant.

• ##### Location, Scale and LOTUS
Joseph Blitzstein

Normal distribution, standardization, LOTUS.

• ##### Midterm Review
Joseph Blitzstein

Midterm review, extra examples.

• ##### Exponential Distribution
Joseph Blitzstein

Exponential distribution, memoryless property.

• ##### Moment Generating Functions
Joseph Blitzstein

Moment generating functions(MGFs), hybrid Bayes' rule, Laplace's rule of sucession.

• ##### Moment Generating Functions Continued
Joseph Blitzstein

MGFs to get moments of Expo and Normal, sums of Poissons, joint distributions.

• ##### Joint, Conditional, and Marginal Distributions
Joseph Blitzstein

Joint, conditional, and marginal distributions, 2-D LOTUS, expected distance between Uniforms, chicken-egg.

• ##### Multinominal and Caucchy
Joseph Blitzstein

Expected distance between Normals, Multinomial, Cauchy.

• ##### Covariance and Correlation
Joseph Blitzstein

Covariance, correlation, variance of a sum, variance of Hypergeometric.

• ##### Transformations and Convolutions
Joseph Blitzstein

Transformations, LogNormal, convolutions, proving existence.

• ##### Beta Distribution
Joseph Blitzstein

Beta distribution, Bayes' billards, finance preview and examples.

• ##### Gamma Distribution and Poisson Process
Joseph Blitzstein

Gamma distribution, Poisson processes.

• ##### Order Statistics and Conditional Expectation
Joseph Blitzstein

Beta-Gamma(bank-post office), order statistics, conditional expectation, two envelope paradox.

• ##### Conditional Expectation Continued
Joseph Blitzstein

Two envelope paradox(cont.), conditional expectation(cont.), waiting for HT vs. waiting for HH.

• ##### Conditional Expectation Given an R.V.
Joseph Blitzstein

Conditional expectation(cont.), taking out what's known, Adam's law, Eve's law, projection picture.

• ##### Inequalities
Joseph Blitzstein

Sum of random numbers of random variables, inequalities(Cauchy-Schwarz, Jensen, Markov, Chebyshev).

• ##### Law of Large Numbers and Central Limit Theorem
Joseph Blitzstein

Law of large numbers, central limit theorem.

• ##### Chi-Square, Student-t, Multivariate Normal
Joseph Blitzstein

Chi-Square, Student-t, Multivariate Normal.

• ##### Markov Chains

Markov chains, transition matrix, stationery distribution.

• ##### Markov Chains Continued
Joseph Blitzstein

Markov chains(cont.), irreducibility, recurrence, transience, reversibility, random walk on an undirected network.

• ##### Markov Chains Continued Futher
Joseph Blitzstein

Markov chains(cont.), Google PageRank as a Markov chain.