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Applied Mathematics and Statistics 553.420/620

Probability

Spring 2024 (4 credits, EQ)

COURSE DESCRIPTION (from the course catalog)

Probability and its applications, at the calculus level. Emphasis on techniques of application rather than on rigorous mathematical demonstration. Probability, combinatorial probability, random variables, distribution functions, important probability distributions, independence, conditional probability, moments, joint distributions, covariance and correlation, conditional distributions, conditional expectation, conditional variance, ordered statistics, exchangeability, inequalities, limit theorems. Students initiating graduate work in probability or statistics should enrollin EN.550.620. Auditors are not permitted.

Recommended Course Background: one year of calculus and mathematical maturity; Co-requisite: multivariable calculus.

Prerequisites

(AS.110.108 [+] AND AS.110.109 [+]) OR (AS.110.106 [+] AND AS.110.107 [+])

Statistics Sequence restriction: students who have already completed EN.550.430 [+] may not register

REFERENCES

My personal course notes will be provided.

(Not required) Sheldon Ross, A First Course in Probability, Pearson Publishing, 9th  edition or later. My personal course notes were written around the 9th  edition of this textbook.  The problems and theoretical  exercises are very good in this textbook (some have answers in the back of the book) and each chapter has a chapter test with full solutions in the back of the book.

Gabe Gormezano, Intuition to Probability**, book (in progress).   Written by one of our undergraduates a PDF version of this book will be posted to Canvas for your enjoyment.   It is an excellent source for problems/exercises to expand your knowledge of the subject.

COURSE ASSESSMENT & GRADE ASSIGNMENT

Weekly homework assignments (25%) – expect a homework assignment* each week without a midterm

*some homework will be posted but will not be collected for grade,

Two (2) midterms (25% each), last exam – noncumulative (25%)

Cut-offs for letter grades to be posted after the grading of each midterm and after last homework.

IMPORTANT DATES

Thursday, FEB. 29, Midterm 1 @ 6:00 in a room to be announced.

Thursday, APR. 04, Midterm 2 @ 6:00 in a room to be announced.

Last exam: MAY 14 @ 2:00-3:30. Location to be announced.

These exam dates are mandatory.  Plan to take these midterms at these times.   If you happen to miss a    midterm, you may be eligible for an incomplete in this course, which you can plan to make-up in January. A university verifiable excuse with valid documentation may be necessary.

OPPORTUNITIES FOR BONUS POINTS

Below I list ways that students can be awarded additional points this semester.   The actual points awarded have not been fully determined or are not being disclosed.  I consider these as incentives, and no one will  be punished for not taking advantage of these opportunities.

.     Homework solutions submitted/typeset completely in LaTeX may receive additional points per assignment. Students will not be deducted points for not submitting LaTeX’d solutions.

.     On exams, there is sometimes a bonus part to a question that will grant a few extra points if answered correctly.

.    By attending and participating in your assigned section, you may receive a bonus to your final

grade if you have perfect/near perfect attendance and participate substantially by answering and asking many questions. If you have perfect/near perfect attendance, but participate more

inconsistently, you may receive a small bonus to your final grade. There is no penalty for a lack of attendance/participation.

EXTRA CREDIT OPPORTUNITIES

Other than the methods listed in the section above, there are no other opportunities to gain extra credit. Please do not ask myself or any TAs for such opportunities.

MIDTERMS

.     To the extent possible midterm examinations will be non-cumulative; however, since this course  builds on itself and introduces definitions and ideas continually, you may be required to recall an idea from an earlier part of the course on midterm examinations.

.     Midterm content will either be made clear in Lectures, Canvas, or by email communication – see also the tentative schedule on last page of this syllabus.   Generally, homework assignments, in-class examples, related book examples/problems, lecture content, old midterms/problems, exercises from recitation sections are all fair-game.

.    Exams will be administered IN-PERSON.   Students will have a fixed amount of time to complete  the exam (typically 75 minutes).  If you’re allowed extended time for midterms, I need you to send me this documentation immediately.

ASSIGNMENTS: SUBMISSIONS & PENALTIES

.    Assignments will be posted in Canvas but collected in Canvas Gradescope.

.    You are to prepare solutions NEATLY to each assignment on your own (no copying others’ work), UPLOAD your solutions (PDF, JPEG, etc.) electronically to the Gradescope menu item   in

Canvas by the deadline (in Baltimore time)!

Also, Gradescope will ask you to tag the pages on which each problem appears in your submission. Be aware: You may lose points in problem numbers that are not tagged.

.    Each assignment is worth  100% and has a deadline (typically by Friday at  11:59PM Baltimore

time of the following week).  You are allowed unlimited attempts at uploading, but any submission will overwrite the previous.   So, make sure you submit your solutions in their entirety!

.     Students are allowed one late submission for the semester (‘late’ means within 24 hours after the

original deadline).  Subsequent late assignments are deducted 20%.

.     Submissions MUST be uploaded to Gradescope; TAs may help you in this process.

.     Students will NOT be able to submit their solutions after the late period and those assignments will receive 0%.  Read: it is better to submit an imperfect/partial assignment than nothing at all.

.     If you are having technical difficulties uploading your assignment, a TA should be able to help you submit it as long as they are given ample time.  Bottom line: do not wait until the last hour to

submit your homework.

.     After uploading your PDF, the student should review their upload to see that all pages have

uploaded properly, completely, in focus, dark and legible and be sure to tag problem numbers to pages within the document.

.    IMPORTANT: You are required to clearly show and explain your work on all HW problems unless otherwise noted in the problem. Solutions with insufficient reasoning or justification are liable to lose points.

ETHICS

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments,

improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded     assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

In addition, I allow (and encourage) collaboration on homework assignments with your friends, colleagues and/or teaching assistants. This is an important part of your learning. However, whatever work you

collaborate on must be thrown away before you write up the corresponding solutions – this is to assure that the work you submit is your own.

Report any violations you witness to the instructor.  You can find more information about university misconduct policies on the web at this site:

For undergraduates: http://e-catalog.jhu.edu/undergrad-students/student-life-policies/

STUDENTS WITH DISABILITIES

Students with disabilities may need accommodations, but first, they must make themselves known to Student Disability Services at Homewood Campus. This office is also available to consult with faculty about any issues or concerns.

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming,  equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions, and temporary disabilities) can request

accommodations for this course by providing an Accommodation Letter issued by Student Disability    Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Homewood Campus, Shaffer Hall #101, call: 410-516-4720 and email:

[email protected] or visit the website.

SOME KEYS TO SUCCESS IN THIS COURSE

This course can be very challenging for most.  It may require a lot of dedication, time and effort, and, quite possibly, some will not be able to manage the workload of this course with their typical

academic/extracurricular schedule.  The following tips I hope will ease the burden of learning = being successful. This list is certainly not comprehensive.

.     Stay on-top of the material. Do NOT let yourself get behind in this course. Do NOT let yourself    pretend you understand what you’ve heard until you are able to apply what you learned in solving problems correctly and see its uses in more abstract situations.  A surprising number of students think they understand a seemingly easy result in this course, but then are not able to see that this    same result was used in a slightly more abstract setting.

.     Do and re-do problems.   There will be many exercises/problems presented to you in this course – through homework, book, via lectures/discussion section – do and re-do them!   If you can’t do

them...

.    Ask questions (of yourself, of others).   Don’t do this course in a vacuum.

.     If you don’t understand why a solution is the way it is, try to find an alternate way to do it – not a futile task as there are often several ways of tackling probability problems.   At the very least, if

you fail at getting the same answer an alternate way, it may help you better understand why the

solution was presented the way it was.  Moral: sometimes failure is the best teacher, but failure can only teach if you can learn from it.

TENTATIVE SCHEDULE

Week 1

Jan-22, lec-01: Experiments, sample spaces, sample points, events, equally likely outcomes and classical probability, basic counting principle, counting ordered n-tuples, sampling with and without replacement .

Jan-24, lec-02: permutations, exchangeability property, anagrams .

Jan-26, lec-03: combinations and binomial coefficients, binomial theorem, properties .

Week 2

Jan-29, lec-04: unordered partitions of fixed sizes , anagrams (revisited), assignments, multinomial coefficients, multinomial theorem.

Jan-31, lec-05: multisets, stars-and-bars counting, number of integer solutions: ordered partitions of an integer.

Feb-02, lec-06: Generalities: sample spaces, events, axioms of probability, consequences of axioms, discrete probability measures .

Week 3

Feb-05, lec-07: conditional probability .

Feb-07, lec-08: conditional probability (continued), law of total probability

Feb-09, lec-09: law of total probability, Bayes rule .

Week 4

Feb-12, lec-10: independent events, conditional independence .

Feb-14, lec-11: random variables (rvs), discrete rvs and their distributions (probability mass functions PMFs), Common discrete distributions: Bernoulli, binomial, hypergeometric .

Feb-16, lec-12: Common discrete distributions (continued): Poisson, geometric, negative binomial .

Week 5

Feb-19, lec-13: Expectation/expected value of a discrete rv, interpretations, expected value of common distributions . Feb-21, lec-14: Expectations of common distributions (continued) . Law of the Unconscious Statistician (LOTUS)

Feb-23, lec-15: Moments of a distribution (or rv), variance of a random variable, higher moment quantities .

Week 6

Feb-26, lec-16: Moment generating function (MGF, Part 1) .

Feb-28, lec-17: RVs in general and their cumulative distribution functions (CDFs), continuous rvs .

Mar-01, lec-18: Common continuous distributions (probability density functions PDFs): Uniform and exponential, Expected value, moments of continuous rvs .

Week 7

Mar-04, lec-19: MGF (part 2), other common continuous distributions: the Gamma distribution .

Mar-06, lec-20: The Gamma distribution (continued), the Normal (Gaussian) distribution .

Mar-08, lec-21: The Normal distribution and its properties .

Week 8

Mar-11, lec-22: Transformations of continuous random variables (The CDF method) .

Mar-13, lec-23: Joint distributions, jointly discrete rvs and their joint PMFs .

Mar-15, lec-24: Jointly continuous rvs and their joint PDFs . Marginal distributions, and independence of rvs .

Week 9

Mar-25, lec-25: The distribution of sums of independent rvs, convolution formulas, discrete convolution . Mar-27, lec-26: Continuous rv case, convolution integral .

Mar-29, lec-27: Conditional distributions discrete case .

Week 10

Apr-01, lec-28: Conditional distributions – continuous case

Apr-03, lec-29: Transformations of continuous random variables (The Method of Jacobians)

Apr-05, lec-30: Conditional distribution mixed type, examples .

Week 11

Apr-08, lec-31: Order statistics (part 1) .

Apr-10, lec-32: Order statistics (part 2) .

Apr-12, lec-33: Exchangeability and exchangeable random variables .

Week 12

Apr-15, lec-34: Covariance, variance of sums, correlation .

Apr-17, lec-35: Conditional expectation, properties, and Law of total expectation .

Apr-19, lec-36: The bivariate Normal distribution .

Week 13

Apr-22, lec-37: Inequalities: Markov, Chebyshev/Jensen, Weak Law of Large numbers (WLLN), Strong law of Large numbers (?) . Apr-24, lec-38: Central limit theorem (part 1) .

Apr-26, lec-39: Central limit theorem (part 2) and closing remarks .