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ECE313

ECE313 (Probability with Engineering Applications) is a 3-credit-hour course that is specifically required for all ECE majors as part of the Technical Core. STAT410 may substitute this course. ECE313 is offered in the fall, spring, and summer.

Content Covered

Foundations

  • Probability Spaces
  • Conditional Probability and Independence
  • Counting

Discrete Random Variables

  • Bernoulli, Binomial, Geometric, Poisson, Negative Binomial Distributions
  • Maximum Likelihood Parameter Estimation
  • Binary Hypothesis Testing
  • Confidence Intervals
  • Markov and Chebyshev Inequalities

Continuous Random Variables

  • Uniform, Exponential, Gaussian
  • Functions of Random Variables
  • Failure Rate Functions

Joint Random Variables

  • Correlation, Covariance
  • Functions of Joint Random Variables
  • Sums of Random Variables (convolution)
  • Minimum Mean Square Error Estimation
  • Law of Large Numbers, Central Limit Theorem
  • Joint Gaussian Distributions

The course is roughly split into four sections. The first section, usually regarded as the easiest, deals with the fundamentals of probability (going all the way down to the axioms of probability) and counting. The second section introduces random variables by starting with the discrete version. It then covers concepts related to random variables in general, such as mean, variance, independence, conditional probabilities, maximum likelihood, confidence intervals, and binary hypothesis testing. The third part of the course then moves towards continuous random variables, and also covers all of the aforementioned topics, but now through the context of continuous random variables instead of discrete random variables. The final part of the course, infamous for its difficulty, covers joint random variables, along with new topics such as minimum mean square error estimation and the central limit theorem. Each portion of the course builds upon the previous portion.

Prerequisites

The official prerequisite to the course is exactly one of MATH257 or MATH416. However, neither course is crucial for any content in ECE313. ECE313 lists these as prerequisites because these courses teach students how to deal with difficult mathematical topics—but as ECE majors, students should be getting practice with this every semester anyway. The course is extraordinarily self-contained - much of the prerequisite material required is covered in the Calculus sequence, such as multivariable integrals, summation of geometric series, Taylor series. Everything is built from the axioms, which are taught at the very beginning of the semester.

When to Take it

It is heavily advised to take this course after ECE210, since this course uses convolution. ECE313 will reach convolution before ECE210, hence taking these concurrently will not help either. While ECE313 does fully introduce convolution, the introduction is extremely rushed.

Many CEs choose to delay this course until their final semester, as it does not open up many upper-level computer engineering courses. While this may seem like a wise decision in order to avoid a negative impact on your GPA, be warned that the likelihood of failing this class is not zero, especially during your final semester when you are mentally checked out.

Electrical engineers are advised to take this course the semester after they take ECE210, as many upper-level electrical engineering courses in areas such as signal processing, communications, controls, and remote sensing require ECE313 as a prerequisite (and for a good reason). This course is also extremely important for those interested in theoretical computer science or machine learning.

Course Structure

During the semester, ECE313 assigns 12-13 written homeworks, due weekly, and 12 PrairieLearn homeworks (with infinite tries), due weekly. There are two midterms and one final.

Many students find the start of ECE313 light, since the axioms probability are often facts they've seen before. However, it will continually ramp up throughout the semester, with each topic building upon the previous one. Do not let this course blindside you: it can be extremely dangerous for your grade.

The homework sets are generally not too difficult and will contain calculations and fun scenarios. Students have found that each week's homework can take very different amounts of time, from under an hour to upwards of five hours to complete. PrairieLearn homeworks are on the easier side - these used to be assigned as weekly 25-minute quizzes with two attempts for each problem, so as homeworks these should be easier and more manageable.

The course will have two midterms and a final, which each exam ramping up from the previous one. There are usually a large bank of past practice exams. However, probability can be notoriously tricky: the questions on the actual exam can look nothing like past practice exams. Students are allowed to bring one double-sided 8.5x11 cheatsheet to the first exam, two double-sided 8.5x11 cheatsheets to the second exam, and so on. Students are highly recommended to reuse cheatsheets from prior midterms, but this is not enforced.

As of Spring 2023, the final exam has not had an average above 50% for four semesters in a row. However, the course is heavily curved, evaluating students based off of a generous objective criteria (hard cutoffs), as well as statistical criteria (class ranking), and giving the better of the two letter grades.

Instructors

ECE313 is usually directed by Professor Hajek, who wrote the course notes. During the semester, lecturers have included Profs. Chen, Shomorony, Do, Zhao, Katselis, and Moharrami. Professor Alvarez usually teaches this course during the summer. Lecturers tend to have backgrounds in related fields including communciations and signal processing.

Course Tips

There is a course textbook written by Bruce Hajek, which also contains many practice problems. The lectures often fall behind and are forced to skip content in order to catch up to other sections, thus it is highly recommended to read the course textbook. Many students have gotten As by only reading the textbook and never going to lecture. It is a great textbook.

Probability is known for never following intuition. Be careful!

The course moves very fast, as it has to cram a lot of content into very little time. The weekly homework is your opportunity to really sit down and learn the material, so it is recommended to do them fully on your own and only go to office hours if you have been stuck for more than an hour. The office hours right before the homework is due are often crowded - instead, attempt the homework early and go to the office hours right after homework is due. You'll probably be able to get a professor to yourself.

Make sure you show your work on exams for some partial credit.

Life After

Probability theory forms the basis of communications, control systems, theoretical computer science, and areas of other fields within electrical and computer engineering. Not surprisingly, ECE313 is a prerequisite for upper level technical electives in many of these areas, such as ECE361 - Digital Communications, ECE418 - Introduction to Image and Video Processing, ECE438 - Communication Networks, ECE459 - Communications Systems, ECE449 - Maching Learning, etc. For students seeking to broaden their intellectual (and possibly career) horizons, probability theory is applicable to modeling in many areas such as finance and economics. If you wish to pursue graduate studies in the aforementioned disciplines, you should consider MATH347 - Fundamentals of Math, MATH444 - Elementary Real Analysis, and MATH447 - Real Variables.

Infamous Topics

  • Functions of random variables: The math can get pretty nasty. However, all this really boils down to is managing inequalities. Make sure you know how to manage inequalities.
  • Joint random variables: It can get confusing as there are a lot of variables going on. It is recommended to develop your own personal system of variables (for example, reserve u and v for the original coordinate system only) and stick to this system for the entire semester.
  • Minimum Mean Square Error Estimation: There are a lot of symbols everywhere. Write down each case (constant, linear, free) on your cheatsheet, and forget about it until the problem on the exam asks for it.

Resources

Professor Hajek's textbook is awesome. The pdf should be linked on the course website and can also be found here. You can learn the entire course from it. Use it.