525.614.82 - Probability & Stochastic Processes for Engineers

Electrical and Computer Engineering
Spring 2024


This course provides a foundation in the theory and applications of probability and stochastic processes and an understanding of the mathematical techniques relating to random processes in the areas of signal processing, detection, estimation, and communication. Topics include the axioms of probability, random variables, and distribution functions; functions and sequences of random variables; stochastic processes; and representations of random processes. Prerequisite(s): A working knowledge of multi-variable calculus, Fourier transforms, and linear systems theory.


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Amit Banerjee


Course Structure

The course materials are divided into modules which can be accessed by clicking Modules on the left 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. 

Course Topics

Here is a summary of the course modules:

Module 1: Concept of Probability
Module 2: Repeated Trials and Random Functions
Module 3: Functions of a Random Variable and Statistical Moments
Module 4: Bivariate Random Variables
Module 5: Two Functions of Two Random Variables and Joint Moments
Module 6: Joint Characteristics, Conditional Distributions and Expected Values
Module 7: Midterm Exam
Module 8: Estimation Techniques and the Law of Large Numbers
Module 9: Hypothesis Testing and Detection Theory
Module 10: Stochastic Processes
Module 11: Stochastic Processes and Linear Systems
Module 12: Power Spectral Estimation
Module 13: Remote Estimation of the Temperature and Size of an Unresolved Object in Space
Module 14: Final Exam

Course Goals

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

  1. Explain theory and basic concepts of probability and stochastic process.
  2. Develop an intuitive and working understanding of probability theory and stochastic process.
  3. Apply probability theory and stochastic process to real-world applications.

Course Learning Outcomes (CLOs)


This course will use:

Papoulis, A. and Pillai, U. (2002). Probability, Random Variables and Stochastic Processes, 4th Ed. McGraw Hill Higher Education, ISBN 978-0-07-3660110. Provided electronically.

Pishro-Nik, H. (2014). Introduction to Probability, Statistics, and Random Process. Kappa Research LLC. ISBN 978-0-9906372-0-2. Can be purchased or use provided text in eReserves.

Stoica, P. Randolf, M. (2005). Spectral Analysis of Signals. Prentice Hall, Upper Saddle River, New Jersey 07458.

Readings from Papoulis and Stoica are available electronically through the EReserves link in Canvas.

Student will need to purchase the Pishro-Nik textbook. You should be able to find it online for about $40. It is a primary text for this course.

Other Materials & Online Resources

eReserves in Canvas contains the Papoulis and Stoica texts.

Required Software

Webcam or microphone are required.  We will also be using MATLAB which is available for free to all JHU students. Here is an online article about how to access MATLAB.

Student Coursework Requirements

It is expected that each module will take approximately 7-10 hours per week to complete. Here is an approximate breakdown: complete the assigned readings and watch video lectures videos (3-4 hours /week); complete assignments including problems and MATLAB exercises (3-5 hours/week); attend weekly office hours, where attendance is optional.

This course consists of 3 requirements:

MATLAB Exercises (15% of final grade)

Twelve total (all modules except 7 and 14): to be completed individually. These activities are designed to provide practice applying the computational concepts you are learning.

Weekly Assignments (35% of final grade)

Twelve total (all modules except 7 and 14): to be completed individually. Assignments consist of problem sets.


Midterm and Final (50% of final grade (25% each))

Both midterm and final exams will be available in Canvas on Wednesday of exam week. Like homework, it is due the following Sunday at midnight, via upload to Canvas. This provides you with five days to independently complete each exam. On Monday and Tuesday of exam weeks instructor will offer one or more office hours to provide pre-exam review. Both exams will be comprehensive in that they will include problems that span all course material covered until that point.

Ungraded practice problems

In addition to the above, the instructor will often provide a set of ungraded practice problems.  These problems do not count toward your grade and are made available if you would like more practice in the content than is provided in the weekly graded problem sets. Frequently, we may discuss these problems during weekly office hours, however, you are welcome to bring questions to office hours related to graded assignments as well. While this may seem onerous, experience has shown that students welcome the availability of additional problems to help them learn and they are of course optional.

Grading Policy

Assignments are due according to the dates posted in your Canvas course site. You may check these due dates in the Course Calendar or the Assignments in the corresponding modules. I/We will post grades one week after assignment due dates.

We generally do not directly grade spelling and grammar. However, egregious violations of the rules of the English language will be noted without comment. Consistently poor performance in either spelling or grammar is taken as an indication of poor written communication ability that may detract from your grade.

A grade of A indicates achievement of consistent excellence and distinction throughout the course—that is, conspicuous excellence in all aspects of assignments and discussion in every week.

A grade of B indicates work that meets all course requirements on a level appropriate for graduate academic work. These criteria apply to both undergraduates and graduate students taking the course.

EP uses a +/- grading system (see “Grading System”, Graduate Programs catalog, p. 10).

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

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.