This course introduces students to the fundamental concepts, methods, and models of applied probability and stochastic processes. The course is application-oriented and focuses on the development of probabilistic thinking and an intuitive feel for the subject rather than on a more formal approach based on measure theory. The main goal is to equip applied mathematics, science, and engineering students with necessary probabilistic tools they can use in future studies and research.

Topics covered include deterministic vs probabilistic models; random variables; joint and marginal distributions; independence; moment generating functions, Markov’s and Chebyshev’s inequalities; law of large numbers, central limit theorem, Monte Carlo method, conditional distributions, conditional expectation and variance; law of total expectation; compound random variables; computing probabilities by conditioning; the best prize problem; the ballot problem; random vectors and matrices; covariance matrix; Karhunen–Loève expansion; transformation of random vectors; Wiener filters; Gaussian random vectors; stochastic processes; counting processes; Poisson processes; interarrival and waiting times, generating the Poisson process; merging and splitting Poisson processes; order statistics; multi-type Poisson process; Brownian motion; hitting times; stationary processes, correlation function, Gaussian processes; power spectral density; and applications to the analysis of the quicksort algorithm, e-commerce, insurance, and health care.

• Ma 3 or EE 55 (enforced prerequisite).
• Some familiarity with MATLAB, e.g. ACM 11, is desired.

Each lecture will be accompanied by two practice problems: a somewhat easier, more practical Problem A, and a more difficult, more conceptual Problem B. The main goal of the practice problems is threefold: to help you better understand the material covered in the corresponding lecture, to help you prepare to solve problems in problem sets and exams, and to accommodate the diversity of students’ math backgrounds by providing both easier and more challenging problems. These problems are for self-practice: they will not be graded, and the solutions (posted on Piazza) also illustrate the expected level of rigor for problem sets and exams.

There will be six Problem Sets. Problems (and solutions) will be posted on Piazza. For assignment and due dates see “Important Dates” below. Late submissions will not be accepted for any reason, but the Problem Set with the lowest score will be dropped and not counted toward your total score. Submitting wrong files or files in a wrong format is considered as a late submission. Extensions may be granted for academic, personal, or medical reasons. For extensions, please email the Head TA.

Your final grade will be based on your total score. Your total score is a weighted average of Problem Sets (50%), Midterm Exam (10%), and Final Exam (40%). You can increase your total score by up to 5% if you participate actively in Piazza discussions in the Q&A section. Each student answer that is submitted before a TA or the instructor posts an answer, and that is later endorsed as a ‘good answer’ by a TA or the instructor, adds 1% to your total score. There are no fixed thresholds for grades, but if your total score is 90% (80%, 70%, 60%), then you are guaranteed at least A (B, C, D).