probability homework solutions

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Course info, instructors.

• Dr. Jeremy Orloff
• Dr. Jennifer French Kamrin

Departments

• Mathematics

As Taught In

• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Introduction to probability and statistics, exams with solutions.

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Lecture notes

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Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

BUS204: Business Statistics

Probability Homework

Solve these problems, then check your answers against the given solutions.

Solutions to Exercises

• {Gl, G2, G3, G4, G5, Yl, Y2, Y3}
• {(HHH) , (HHT) , (HTH) , (HTT) , (THH) , (THT) , (TTH) , (TTT)}
•  P(H or G) = P(H) + P(G) - P(H and G) = 0.26 + 0.43 - 0.14 = 0.55
• P( NOT (H and G) ) = 1 - P(H and G) = 1 - 0.14 = 0.86
• P( NOT (H or G) ) = 1 - P(H or G) = 1 - 0.55 = 0.45
• P (Type O or Rh-) = P(Type O) + P(Rh-) - P(Type O and Rh-) 0.52 = 0.43 + 0.15 - P(Type O and Rh-); solve to find P(Type O and Rh-) = 0.06 6% of people have type O Rh— blood
• P( NOT (Type O and Rh-) ) = 1 - P(Type O and Rh-) = 1 - 0.06 = 0.94 94% of people do not have type O Rh— blood
• P(R or F) = P(R) + P(F) - P(R and F) = 0.72 + 0.46 - 0.32 = 0.86
• P( Neither R nor F ) = 1 - P(R or F) = 1 - 0.86 = 0.14
• P(D and E) = P(D I E)P(E) = (0.20)(0.40) = 0.08
• P(E I D) = P(D and E) / P(D) = 0.08/0.10 = 0.80
• P(D or E) = P(D) + P(E) - P(D and E) = 0.10 + 0.40 - 0.08 = 0.42
• Not Independent: P(D I E) = 0.20 which does not equal P(D) = .10
• Not Mutually Exclusive: P(D and E) = 0.08 ; if they were mutually exclusive then we would need to have P(D and E) = 0, which is not true here.

Statistics 110: Probability

A free online version of the second edition of the book based on Stat 110,  Introduction to Probability  by Joe Blitzstein and Jessica Hwang, is now available at  http://probabilitybook.net

Print copies are available via  CRC Press , Amazon , and elsewhere. 

Stat110x is also available as an edX course. Free signup at  https://www.edx.org/course/introduction-to-probability-0

The edX course focuses on animations, interactive features, readings, and problem-solving, and is complementary to the Stat 110 lecture videos on YouTube, which are available at  https://goo.gl/i7njSb

The Stat110x animations are available within the course and at  https://goo.gl/g7pqTo

• https://twitter.com/stat110
• https://www.youtube.com/playlist?list=PLLVplP8OIVc8EktkrD3Q8td0GmId7DjW0

stat110picture.png

Fall 2020, Lecture B00 (Kemp) TR 12:30-1:50pm

Probability theory i.

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Announcements

Last Modified: 9/23/15

• Welcome : to Math 280A: Probability Theory I, in Fall 2020!
• Quiz 1 grades have been released on Gradescope. The regrade request window is now close.
• Quiz 2 grades have been released on Gradescope. The regrade request window is Monday 10/26/20 8am to Wednesday 10/28/20 at 8pm . If you wish to lodge a regrade request on Quiz 1, please do so during this period of time.

Course Information

• Textbook : The main source we will follow are Bruce Driver's excellent Probability notes: Probability Tools with Examples , by Bruce Driver Here are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel. Probability: Theory and Examples (5th Edition), by Rick Durrett A Probability Path by Sidney Resnick A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray Probability Theory: A Comprehensive Course by Achim Klenke If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here .
• Coursework : There will be weekly homework assignments due on Mondays (starting in Week 2); they are posted below . There will be 5 quizzes , in weeks 1, 3, 5, 7, and 9 of the quarter; they will take place during the scheduled Thursday lecture time, with an alternate sitting available in the late evening to accommodate those in distant time-zones. And there will be a take-home final exam during exam week. Timing and due dates for all courses assessments can be found below .
• Piazza is an online discussion forum. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, quizzes, etc. The instructor and TA will also monitor and post to Piazza regularly. You can sign up here . Note: Piazza has an opt-in "Piazza Careers" section which, if you give permission, will share statistics about your Piazza use with potential future employers. It also has a "social network" component, based on other students who've shared a Piazza-based class with you, that comes with the usual warnings about privacy concerns. Piazza is fully FERPA compliant, and is an allowed resource at UCSD. Nevertheless, you are not required to use Piazza if you do not wish.
• Gradescope is an online tool for uploading and grading assignments and exams (it is now under the umbrella of Turnitin). You will turn in your homework, quizzes, and final exam through Gradescope, and you will access your graded assessments there as well. Access the class Gradescope site here .

Instructional Staff

We will be communicating with you and making announcements through an online question and answer platform called Piazza (sign up link: piazza.com/ucsd/winter2016/math180a ). We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response. Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework through Piazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors, in office hours. You can also post private messages to instructors on Piazza, which we prefer to email.

Our office hours, and all relevant scheduled course activities, can be found in the following calendar.

Recorded Lectures

The Lectures for this course are pre-recorded, and available on YouTube .

The lectures are not divided into even 80-minute chunks. They are organized by topic, concept, or example.

Below, you will find a list (with links) of the lecture videos you should watch prior to the listed date, along with pdf slides of the tablet output during those lecture videos.

Math 280A is the first quarter of a three-quarter graduate level sequence in the theory of probability. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability. A strong background in undergraduate real analysis at the level of Math 140AB is essential for success in Math 280A. In particular, students should be comfortable with notions such as countable and uncountable sets, limsup and liminf, and open, closed, and compact sets, and should be proficient at writing rigorous epsilon-delta style proofs. Graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter course in stochastic processes which will be offered in Winter 2021. See also this page , maintained by Ruth Williams, for more information on graduate courses in probability at UCSD.

According to the UC San Diego Course Catalog , the topics covered in the full-year sequence 280ABC include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, convergence in distribution, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. Given the current pandemic crisis and emergency remote teaching modality, it is more difficult than usual to predict what pace we will work through this material, and where the dividing line between 280A and 280B will occur.

Prerequisite:   Students should have mastered the fundamentals of real analysis in metric spaces, as covered in MATH 140AB, before taking this course. An undergraduate course in probability, comparable to MATH 180A, and further courses in stochastic processes, comparable to MATH 180BC, would also be an asset, but are not absolutely necessary.

Lectures:   The lectures for this course will be recorded asynchronously, and made available on YouTube. You should engage with the relevant videos before each "Lecture" session. The schedule Lecture times will be devoted to Q&A sessions and quizzes. The Q&A sessions will be recorded, with recordings available on Canvas; the quizzes will not be recorded (but will take place live on Zoom), and will be available in a "second sitting" to accommodate those students in far-flung time-zones.

Homework:   Homework assignments are posted below , and will be due by 9pm (with a 30-minutes "late" grace period in case of technical glitches) on Mondays throughout thee quarter. You must turn in your homework through Gradescope; if you have produced it on paper, you can scan it or simply take clear photos of it to upload. You must select pages corresponding to your solutions of problems during the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you have not selected pages when the TA begins grading, the TA will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Quizzes:   There will be 5 quizzes throughout the quarter, to test your fundamental knowledge of the course material. You will write them on Thursdays 1-1:50pm or 7-7:50pm, live on Zoom (so that your instructional team can answer questions if any arise), and turn them in via Gradescope. No collaboration (with other humans or with online resources) is allowed on quizzes.

Lowest scores:    Of the 9 homework assignments, only your highest 7 scores will count towards your final grade. Of the 5 quizzes, only your highest 4 scores will count towards your final grade.

Final Exam:   The final exam will be take-home. It will be available and due during exam week; more details about the exam window will be available later in the term. No collaboration (with other humans or with online resources) is allowed on the final exam. We reserve the right to invite students to follow-up Zoom meetings after the final exam to confirm that the work was completed without collaboration. We reserve the scheduled final exam time-slot for this purpose.

• Exam Responsibilities   An outline of the responsibilities of faculty and students with regard to final exams
• Policies on Examinations   The Academic Senate policy regarding final examinations (These are the rules!)

Regrade Policy:    Your quizzes, homeworks, and final exam will be graded using Gradescope . For quizzes and the final exam , you will be able to request regrades through Gradescope for a specified window of time. Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline. For homework , any clerical erros (such as a problem or page that the TA accidentally missed when grading) should be discussed with the TA during office hours. Grading rubrics are not negotiable ; if the TA has taken off some number of points from your solution, there is a sound pegagogical reason for this. This is a PhD class in mathematics . We are not focused on numerical grades here; we are focused on learning deep and challenging material. The grading is meant as a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the concepts and thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to think and work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you did not receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for example through Piazza), should you consider approaching your TA or instructor for further explanation of grading choices. However, please understand that these conversations will not result in a change in your grade unless there has been some clear clerical error, such as the TA accidentally missing part of your solution. The TA will not change their assessment of a students work due to conversations or complaints after the fact.

• 40% Homework,  30% Quizzes,  30% Final Exam

Academic Integrity:   UC San Diego's code of academic integrity outlines the expected academic honesty of all students and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers and instructors respectfully in all forms of interaction (Zoom meetings, email, Piazza discussions, etc).

Assessment Versioning : following UCSD (and common) practice, recommended by the Academic Integrity Office, assessments given at non-overlapping times will be comparable , but may not be identical . In particular: the two sittings of each Quiz (and potentially quizzes given within each sitting) may not have exactly the same questions, but will be designed to cover the same material and be of equivalent levels of difficulty. This practice is meant to maintain course integrity, avoiding unpermitted collaboration (either intentional or accidental).

OSD Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter will be issued by the OSD electronically. Please make arrangements to discuss your accommodations with me in advance ( by the end of Week 2 ). We will make every effort to arrange for whatever accommodations are stipulated by your AFA letter. For more information, see here .

Covid-19 Accommodations: Due to the unprecedented Covid-19 pandemic, our lectures and all meetings will be remote, using Zoom, this year. Some of you are residing in different time-zones (and different continents), and these present additional challenges. To accommodate, all lectures are pre-recorded asynchronously, and the Q&A sessions are also recorded. Office hours will not be recorded, but will be distributed throughout different days and times so all students should be able to attend in regular work hours; failing that, individual Zoom meetings can be made. Synchronous quizzes will be held in two "sittings": 1-1:50pm and 7-7:50pm Pacific Time, which should cover everyone enrolled in the class, during daylight hours. If these accommodations are still insufficient due to a severe Covid-19 pandemic related issue you have, please contact me no later than Friday, October 16 , to discuss other arrangements.

Weekly homework assignments are posted here. Homework is due by 9:00pm on the posted date, through Gradescope . Late homework will not be accepted.

• Homework 1 , due Monday, October 12.
• Homework 2 , due Monday, October 19.
• Homework 3 , due Monday, October 26.
• Homework 4 , due Monday, November 2.
• Homework 5 , due Monday, November 9.
• Homework 6 , due Monday, November 16.
• Homework 7 , due Monday, November 23.
• Homework 8 , due Monday, November 30.
• Homework 9 , due Monday, December 7.
• Department of Mathematics, UC San Diego

Probability-2

• © 2019
• 3rd edition
• View latest edition
• Albert N. Shiryaev 0

Department of Probability Theory and Mathematical Statistics, Steklov Mathematical Institute and Lomonosov Moscow State University, Moscow, Russia

You can also search for this author in PubMed   Google Scholar

• Along with Probability-1, forms the third English edition of the author’s classic Probability
• Offers new problems, exercises, proofs, and applications in financial topics and mathematical statistics
• Features a Historical Review charting the development of the mathematical theory of probability
• Synthesizes classical ideas and results with many of the major achievements of modern probability theory
• Suitable for a course on random processes or for independent study

Part of the book series: Graduate Texts in Mathematics (GTM, volume 95)

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Table of contents (5 chapters)

Front matter, sequences and sums of independent random variables.

Albert N. Shiryaev

Stationary (Strict Sense) Random Sequences and Ergodic Theory

Stationary (wide sense) random sequences: l 2 -theory, martingales, markov chains, back matter.

• probability theory
• random processes
• discrete time processes
• random sequences
• probability textbook
• random processes textbook
• sums of independent random variables
• stationary random sequences
• martingales
• Markov chains
• zero-one laws
• convergence of series
• strong law of large numbers
• law of the iterated logarithm
• financial mathematics
• financial engineering

About this book

This textbook is the second volume of a pair that presents the latest English edition of the author’s classic, Probability . Building on the foundations established in the preceding Probability-1 , this volume guides the reader on to the theory of random processes. The new edition includes expanded material on financial mathematics and financial engineering; new problems, exercises, and proofs throughout; and a Historical Review charting the development of the mathematical theory of probability. Suitable for an advanced undergraduate or beginning graduate student with a course in probability theory, this volume forms the natural sequel to Probability-1 .

Probability-2 opens with classical results related to sequences and sums of independent random variables, such as the zero–one laws, convergence of series, strong law of large numbers, and the law of the iterated logarithm. The subsequent chapters go on to develop the theory of random processes with discrete time: stationary processes, martingales, and Markov processes. The Historical Review illustrates the growth from intuitive notions of randomness in history through to modern day probability theory and theory of random processes.

Along with its companion volume, this textbook presents a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, the measure-theoretic foundations of probability theory, weak convergence of probability measures, and the central limit theorem. Many examples are discussed in detail, and there are a large number of exercises throughout.

Authors and Affiliations

About the author, bibliographic information.

Book Title : Probability-2

Authors : Albert N. Shiryaev

Translated by : Dmitry M. Chibisov

Series Title : Graduate Texts in Mathematics

DOI : https://doi.org/10.1007/978-0-387-72208-5

Publisher : Springer New York, NY

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer Science+Business Media, LLC, part of Springer Nature 2019

Hardcover ISBN : 978-0-387-72207-8 Published: 25 March 2019

Softcover ISBN : 978-1-0716-1829-5 Published: 21 July 2021

eBook ISBN : 978-0-387-72208-5 Published: 23 March 2019

Series ISSN : 0072-5285

Series E-ISSN : 2197-5612

Edition Number : 3

Number of Pages : X, 348

Number of Illustrations : 16 b/w illustrations

Additional Information : Originally published in one volume; English translation of the 4th original Russian edition published by © Shiryaev A. N., 2007 and © MCCME, 2007

Topics : Probability Theory and Stochastic Processes

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