525.614.8VL - Probability & Stochastic Processes for Engineers

Course Structure

The course materials are divided into modules which can be accessed by clicking Course Modules on the course menu. A module will have several sections including the overview, content, readings, discussions, and assignments. You are encouraged to preview all sections of the module before starting. Most modules run for a period of seven (7) days, exceptions are noted in the Course Outline. You should regularly check the Calendar and Announcements for assignment due dates.

Course Topics

• Definitions of Probability
• Axioms of Probability
• Conditional Probability 
• Bernoulli Random Trials
• Definition of a Random Variable
• Bayes Theorem
• The Law of Total Probability
• Function of a Random Variable
• Synthesis of a Random Variable
• Mean, Variance, Moments
• Characteristic Functions
• Moment Generating Functions
• Joint CDF and PDF
• Marginal Densities
• Jointly Normal Random Variables
• One Function of 2 Random Variables
• Two Functions of 2 Random Variables
• Auxiliary Random Variable
• Joint Moments
• Correlation Coefficient
• Definitions of Uncorrelated, Orthogonal, and Independence
• Joint Characteristic Functions
• Convolution Theorem
• Conditional Distribution
• Conditional Expectation
• Sequences of Random Variables
• Linear Transformations of Random Variables
• On Transformation of Random Variables
• Conditional Densities
• Stochastic Convergence
• Central Limit Theorem
• Stochastic Processes
• First Order Independent Increments
• Weiner Process
• Stationarity
• Ergodicity
• Systems with Stochastic Inputs
• Discrete Markov Chains

Course Goals

By the end of the course, you will be able to:

• Understand basic probability theory and the concept of a random variable
• Master the skills required to characterize a random variable such as the cumulative distribution, probability mass, probability density, and characteristic functions
• Characterize functions of random variables.
• Demonstrate the ability to characterize the joint properties of two or more random variables
• Simulate various types of random variables and basic probability theory concepts
• Characterize a random process and transformations of random processes
• Understand the basic concepts of Discrete Markov Chains

Textbooks

Pishro-Nik, Hossein (2014). Introduction to Probability, Statistics and Random Processes. Kappa Research. Published online at www.probabilitycourse.com (Recommended)

Other Materials & Online Resources

Papouls and Pillai (2002). Probability, Random Variables, & Stochastic Processes (4th Ed.). McGraw-Hill.

Leon-Garcia, Alberto (1994). Probability and Random Processes for Electrical Engineering (2nd Ed.). Addison Wesley Longman.

Kay, Steven (2006). Intuitive Probability and Random Processes using Matlab. Springer.

Ross, Sheldon (2014). Introduction to Probability Models (11th Ed.). Elsevier.

Required Software

We will be having several homework assignments and a course project that will require you to simulate some of the probabilistic concepts you will be learning in class. This course will be taught with an assumption that you will be using MATLAB, but other programming languages such as Python, Mathematica, or Octave are also fine.

You can access a recent version of MATLAB with the latest toolboxes. The MATLAB Total Academic Headcount (TAH) license is now in effect. This license is provided at no cost to you. Send an email to software@jhu.edu to request your license file/code. Please indicate that you need a standalone file/code. You will need to provide your first and last name, as well as your Hopkins email address. You will receive an email from Mathworks with instructions to create a Mathworks account. The MATLAB software will be available for download from the Mathworks site.

Student Coursework Requirements

Students are expected to attend each course lecture which will be on Mondays from 4:30–7:10 PM. To encourage attendance and in-class interaction, there will be a participation grade. Homework assignments will be due at the beginning of each class period. There will also be one in-class midterm exam (or course project), and an in-class final exam.

Course Project and Report

The course project will give you an opportunity to implement some of the probabilistic concepts that you will be learning in this course. The course project is evaluated by the following grading elements:

1. Introduction giving an overview of the chosen topic (10%)
2. Mathematical Formulation of the chosen topic (20%)
3. Simulation of an aspect of the project topic (30%)
4. Discussion of your simulation results and Conclusions (30%)
5. References (10%)

For this project, you will choose a topic that would be beneficial for the work you currently do or would like to do at your place of employment. It is also fine if you choose a topic that matches your personal interests. You will be required to hand in a hard copy of your report and send me an electronic version as well. The electronic submission of your report should include your simulation source code.

Grading Components

[1] Participation (Weekly) - 10%
[2] Homework Assignments (Weekly) - 30%
[3] Midterm OR Course Project - 30%
[4] Final - 30%

Grading Policy

Final grades will be determined by overall class performance, but the following are appropriate guide lines:

100-98 = A+
97-94   = A
93-90   = A-
89-87   = B+
83-86   = B
82-80   = B-
79-77   = C+
76-73   = C
72-70   = C-
69-67   = D+
66-63   = D
<63      = F

Course Policies

Homeworks: Homework will be assigned weekly and is due at the beginning of each class period. No late homework will be accepted without prior approval by the instructor.  You can work with your fellow classmates on homework but there must be no copying and you should write out your own solutions.

Tests: It is mandatory that you take the in-class midterm (or do the course project) and final exam. The exams will be open book, but you will not be allowed to use any online resources except for our Canvas class page.

Make-ups: If you cannot make a test due to a family emergency or a work situation such as required travel, arrangements can be made to take the test on another date.

Other Information: Although not mandatory, it is expected that you will come to class. There is a participation component in your final grade to encourage this. Also, you are encouraged to ask questions during each class period.

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

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 Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.