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Introduction to Probability

Introduction to Probability 15
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ZoeTabbot,Germany,Professional
Published Date:13-07-2017
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Welcome to Stat 110: Introduction to Probability Unit 1: Intro to Probability and Counting 1 Unit 1 Outline • Course logistics and details • What is Probability? • Probability Axioms/Rules • Counting • The Birthday Problem • Story Proofs 2 Stat 110 • Prereq’s: • Multivariable Calculus (equivalent to Math 21a…can be taken concurrently) • The course material is nothing like what you may have seen in an introductory statistics course (like stat 104). The material will be quite mathy and calculus-based, eventually. • It will be example and problem driven, and will not include many theoretical proofs, but will include many heuristic and intuitive mathematical justifications: often referred to as “story proofs”. 3 Kevin’s Contact Info • My office: Science Center, Room SC-614 • Office Hours: • Tues 2:30-3:30pm, and Thurs11:45am-12:45pm • Also happily by appointment (via email) • Phone numbers: • Statistics Department: (617) 495-5496 • My office (SC-614): (617) 495-8711 • Email: kraderfas.harvard.edu (best way to get a hold of me) 4 Teaching Staff • Teaching Fellows: Head TF: Viviana Garcia (vgarciafas.harvard.edu) Other TFs: too many to list. See course website “Teaching Staff and Office Hours” for finalized list by the end of the week: http://isites.harvard.edu/icb/icb.do?keyword=k104821&pageid=icb.page690517 • Teaching fellows will be teaching sections, holding office hours, answering questions via email, and grading HW’s and exams. 6 Course Website Course website: http://isites.harvard.edu/icb/icb.do?keyword=k104821 • There you will find (eventually): • Syllabus • Administrative Announcements • Lecture Notes and Videos • Section Material • Assigned Homeworks • Other Study Material (practice exams, web links, etc...) 7 Lectures • Meeting Times: • Tues & Thurs, 1–2:30pm in Sanders Theater in Memorial Hall. • Lectures are not mandatory. But the easiest way to succeed in this course (and in college) is to attend class and pay attention • Lectures will be videotaped, and the videos will be posted on the course website. • Exams will be based on the lectures and HWs. 8 Sections • Optional (but strongly recommended) weekly section to discuss homework, do extra problems, and review difficult concepts. • Meeting times: all throughout the week (mostly Tuesday through Friday, afternoons and evenings) • Sections will begin next week (Sept 8-12) as the first HW is due Fri, Sept 12. • Look for announcement on the course website for permanent times (OH’s too). 9 Lecture Notes • Paper copies will NOT be handed out at the beginning of lecture after Unit 2 (we will provide copies up til that point). • They’ll be organized unit-by-unit (which will roughly follow the chapters in the text). Each unit is about 3 hours of lecture time (with random variation around that average time). • Future lecture notes will be posted online about 24 hours in advance. An email will be sent when they are posted. • Notes are very concise – you are encouraged to add your own annotations and develop your own notes. • Occasionally mistakes appear in lecture notes; corrected versions will be posted after class. 10 Recommended Textbook (not required) • Introduction to Probability, Blitzstein & Hwang, st 1 edition. Amazon Link: www.amazon.com/Introduction-Probability-Chapman-Statistical-Science/dp/1466575573 Publisher Direct, 20% off as of Aug 27…may now be expired: http://www.crcpress.com/product/isbn/9781466575578 • Some of the assigned homework problems will be assigned from the text, but will always be reproduced for you on the assignment itself. • Exams will be based on the lectures directly, and nothing new from the text not seen in the lectures, notes, or HW’s. • It’s a great reference for more details on what is seen in the lectures. The lectures will follow the text pretty closely. 11 Computing (not required at all) • Statistical Computing Package: R • Can be downloaded from: http://cran.us.r-project.org/ • Rstudio program will be encouraged. http://www.rstudio.com/ • Some demonstrations and calculations will be presented in class, but R will not be used besides that. It is a great tool to learn if you are going to practice statistics in the future. • Some calculations on HW will asked to be simplified numerically. Any of R, google, or an online calculator (or anything else) can be used for these. 12 Exams • Midterm: Thurs, Oct 23, 1-2:30pm (location TBD). • Final Exam: Exam Group 8. Date and location TBD. • You will be given a page of distributions on the exam (see online for the exact sheet). • You will be allowed one sheet of notes for the midterm and two sheets for the final exam. Front-and-back OK. • Exams are difficult. The median is often below 70. 13 Homeworks • Posted to course website one week in advance. There will be 11 of them, and they will be due at 1:10pm the following Friday. • HW 1 will be posted by Friday at noon, and will be th due the next Friday, Sept 12 , at 1:10pm. • Submission: paper copy to the Stat 110 HW collection box (location TBD). Directions will be at the top of the assignment. • No HWs are dropped . You will be allowed one late HW (up to 96 hours later) with no questions asked. You will be allowed to submit late HW with an official excuse (note from UHS or your resident dean). 14 HW Collaboration • You are encouraged to discuss homework with other students (and with the instructor and TAs, of course), but you must write your final answers yourself, in your own words. • Solutions prepared “in committee” or by copying or paraphrasing someone else’s work are not acceptable; your handed-in assignment must represent your own thoughts. • Please indicate on your problem sets the names of the students with whom you worked. 15 Course Grading Component Weighting1 Weighting2 Homework 30% 30% Midterm 20% 35% Final Exam 50% 35% Total 100% 100% Your overall score for the course will be the higher of the 2 weighting schemes presented above. Final course letter grades are not assigned according to a fixed percentages of A's, B's, etc… 16 Course Description A comprehensive introduction to probability, as a language and set of tools for understanding statistics, science, risk, and randomness. Basics: sample spaces and events, conditional probability, and Bayes' theorem. Univariate distributions: density functions, expectation and variance, Normal, t, Binomial, Negative Binomial, Poisson, Beta, and Gamma distributions. Multivariate distributions: joint and conditional distributions, independence, transformations, and Multivariate Normal. Limit laws: law of large numbers, central limit theorem. Markov chains: transition probabilities, stationary distributions, convergence. 17 Unit 1 Outline • Course logistics and details • What is Probability? • Probability Axioms/Rules • Counting • The Birthday Problem • Story Proofs 18 Probability is the basis of Statistics • Statistics is the field that studies uncertainty (while Mathematics studies certainty). Probability is useful to help explain this uncertainty and variation/variability. • Probability is used in many other fields: • Biology: used all the time in Genetics (and with Bayes theorem, which we will study extensively). • Medicine: clinical trials are rooted in probability. • Physics: quantum mechanics is based almost completely on probability (of where an electron is). • Gambling: Historically, probability was first studied based on games of chance. Bring down the house • Finance: “betting” on the stock market’s uncertainty Probability Terminology Terminology • random experiment – an experiment whose individual outcomes are uncertain but there is a regular distribution in a large number of repetitions.  Examples: • Coin tossing and dice rolling • The lottery and other games of chance • Results of taking a drug in a clinical trial. • outcome: the value of one replication of a random experiment. Written as little s.  Coin Tossing: • s = H with one toss of a coin • s = HTT with three tosses 20 Probability Terminology (cont.) • sample space (labeled S): is the set of all possible outcomes of a random experiment • Examples: 1. Toss a coin three times: S = HHH,THH,HTH,…,TTT 2. Face showing when rolling a six-sided die: S = 1,2,3,4,5,6 3. Pick a real number between 1 and 20: S =1,20 4. Sex of a randomly selected person: S = Male, Female • event (labeled A, B, etc…): a set of outcomes of a random experiment. • Examples: 1. The event A that exactly two heads are obtained when a coin is tossed three times: A =HHT,HTH,THH 2. The result of the toss of a fair die is an even number: A = 2,4,6 3. The number chosen from the set of all real numbers between 1 and 20 is at most 8.23: A = 1,8.23 4. A randomly selected person is Female: A = Female 21