Stat 110 Final Review, Fall 2011Prof. Joe Blitzstein1General InformationThe final will be on Thursday 12/15, from 2 PM to 5 PM. No books, notes, computers,cell phones, or calculators are allowed, except that you may bring four pages ofstandard-sized paper (8.5” x 11”) with anything you want written (or typed) onboth sides. There will be approximately 8 problems, equally weighted. The materialcovered will be cumulative since probabilityiscumulative.To study, I recommend solving lots and lots of practice problems!It’s a goodidea to work through as many of the problems on this handout as possible withoutlooking at solutions (and then discussing with others and looking at solutions tocheck your answers and for any problems where you were really stuck), and to takeat least two of the practice finals under timed conditions using only four pages ofnotes. Carefully going through class notes, homeworks, and handouts (especially thishandout and the midterm review handout) is also important, as long as it is doneactively(intermixing reading, thinking, solving problems, and asking questions).2Topics•Combinatorics: multiplication rule, tree diagrams, binomial coefficients, per-mutations and combinations, sampling with/without replacement when orderdoes/doesn’t matter, inclusion-exclusion, story proofs.•Basic Probability: sample spaces, events, axioms of probability, equally likelyoutcomes, inclusion-exclusion, unions, intersections, and complements.•Conditional Probability: definition and meaning, writingP(A1\A2\· · ·\An)as a product, Bayes’ Rule, Law of Total Probability, thinking conditionally,prior vs. posterior probability, independence vs. conditional independence.•Random Variables: definition and interpretations, stories, discrete vs. contin-uous, distributions, CDFs, PMFs, PDFs, MGFs, functions of a r.v., indicatorr.v.s, memorylessness of the Exponential, universality of the Uniform, Poissonapproximation, Poisson processes, Beta as conjugate prior for the Binomial.sums (convolutions), location and scale.1
•Expected Value: linearity, fundamental bridge, variance, standard deviation,covariance, correlation, using expectation to prove existence, LOTUS.•Conditional Expectation: definition and meaning, taking out what’s known,conditional variance, Adam’s Law (iterated expectation), Eve’s Law.•Important Discrete Distributions:Bernoulli, Binomial, Geometric, NegativeBinomial, Hypergeometric, Poisson.•Important Continuous Distributions: Uniform, Normal, Exponential, Gamma,Beta, Chi-Square, Student-t.•Jointly Distributed Random Variables: joint, conditional, and marginal distri-butions, independence, Multinomial, Multivariate Normal, change of variables,order statistics.•Convergence: Law of Large Numbers, Central Limit Theorem.•Inequalities: Cauchy-Schwarz, Markov, Chebyshev, Jensen.