Instructor Information

Stacy Hill

Stacy D. Hill received the B.S. and M.S. degrees from Howard University in 1975 and 1977, respectively, and the D.Sc. in Systems Science and Mathematics from Washington University in St. Louis, in 1983.  Since 1983 he has been on the Senior Professional Staff of the Johns Hopkins University Applied Physics Laboratory.  Dr. Hill is a Research Faculty member of the Applied and Computational Mathematics (ACM) Program and also serves on the program's Advisory Board.

Course Information

Course Description

This course provides a rigorous, measure-theoretic introduction to probability theory. It begins with the notion of fields, sigma fields, and measurable spaces and also surveys elements from integration theory and introduces random variables as measurable functions. It then examines the axioms of probability theory and fundamental concepts including conditioning, conditional probability and expectation, independence, and modes of convergence. Other topics covered include characteristic functions, basic limit theorems (including the weak and strong laws of large numbers), and the central limit theorem.

Prerequisites

EN.625.601 Real Analysis and EN.625.603 Statistical Methods and Data Analysis.

Course Goal

Develop main tools and fundamental results of probability theory from first principles.

Course Objectives

• Develop measure and integration theory concepts used in probability.
• Apply notions from measure and integration to develop basic tools used in probabilistic analysis.
• Master fundamental tools and techniques used in probabilistic analysis.

When This Course is Typically Offered

Spring semester, even years, at the Applied Physics Laboratory

Syllabus

• Probability measures
• Random vectors
• Probability distribution function of a random variable
• Mathematical expectation and its properties
• Conditional expectation
• Stochastic independence
• Characteristic functions
• Modes of convergence for random variables
• Laws of large numbers
• Central Limit Theorem

Student Assessment Criteria

 Homework 30% Midterm Exam 35% Final Exam 35%

Homework will be assigned regularly and will be due one week from the day it is assigned. Occasionally, the instructor may select a subset of the assigned problems to grade completely. However, all problems will be graded for completeness and most will, at a minimum, be graded for technical correctness.

All homework is due as assigned. A late assignment that is no more than one week late will be given half the credit it otherwise would have received.  Homework that is later than one week will be given no credit. Exceptions to the late policy will be made on a case-by-case basis in extreme circumstances. The student must contact the instructor in such cases.

Textbooks

Textbook information for this course is available online through the MBS Direct Virtual Bookstore.

Course Notes

There are no notes for this course.

Final Words from the Instructor

Prior knowledge of measure theory is not assumed; the development of concepts from the theory of measure and integration will be self-contained.

Students are expected to have a working knowledge of the following topics from Real Analysis:

• Completeness property of the real numbers
• Definition of complete ordered field
• The infimum and supremum of a bounded nonempty set of real numbers
• Definition of the of limit of a function, continuous function
• Metric space topology: definition of open set, closed set, compact set; the Heine-Borel and the Bolzano-Weierstrass theorems in (k-dimensional) Euclidean space.
• Definition of the limit inferior and the limit superior of a real-valued sequence of real numbers
• Definition and properties of the Riemann integral.