Sum of random numbers of random variables, inequalities(Cauchy-Schwarz, Jensen, Markov, Chebyshev).
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.
Harvard University, founded in 1636, is America's oldest Ivy League university. Located in Cambridge, Massachusetts, Harvard consists of an undergraduate program, the College, as well as 13 other graduate level schools and institutes offering top-ranked programs in fields such as Medicine, Business, and Law. Notable alumni include former U.S. president Franklin Delano Roosevelt, poet Robert Frost, architect Frank Gehry, songwriter Leonard Bernstein, and comedian Conan O'Brien.