# Probability and Random Processes

Victoria University of Wellington

## Course Description

• ### Course Name

Probability and Random Processes

• ### Host University

Victoria University of Wellington

• ### Location

Wellington, New Zealand

• ### Area of Study

Mathematics, Statistics

• ### Language Level

Taught In English

• ### Prerequisites

MATH 243; MATH 277 or STAT 232

• ### Course Level Recommendations

Upper

ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.

### Hours & Credits

• Credit Points

15
• Recommended U.S. Semester Credits
3 - 4
• Recommended U.S. Quarter Units
4 - 6
• ### Overview

The course provides a firmer foundation in probability theory and an introduction to random processes. Introductory topics: continuity of probability measures; Stieltjes integrals; almost sure convergence. Main topics: conditional distributions and effects of conditioning; martingales in discrete time; Poisson point processes; birth and death processes; renewal processes.

Course content
This is an outline of the contents of the course. We may not be able to cover all these topics. The basic aim is to broaden probabilistic education of students and to introduce them to rich world of random processes.

1. Probability distributions on σ− algebra of events. The particular topics include a. Concept of algebra and σ− algebra of events b. Probability distributions: their continuity and σ− additivity c. Distribution functions, especially those which are neither continuous nor discrete

2. Characteristic convergence of probability theory – almost sure convergence: the concept and examples a. Borel-Cantelli lemma b. Further examples of independent interest

3. Conditioning. We will not restrict the approach to the case when the probability of the condition is positive - it can also be 0. The particular topics include a. Conditional probabilities and conditional distributions b. Conditional expectations. Total probability formula c. Examples and first applications

4. Martingales in discrete time. We consider one of the main class of random processes in probability theory - the martingales. They play fundamental role in wide class of applications. In particular, they form basis for probabilistic modeling in finance. The particular topics include a. Doob’s inequality b. Markov inequality c. Examples of positive martingales. Their asymptotic behaviour. d. Random walk and other examples

5. Random sums and Compound Poisson processes. Very large number of phenomena can be modeled as a compound Poisson process. Risk process of an insurance company is one of them. The particular topics include: a. Random sums and their moments b. Applications to finance and insurance c. Ruin probability

6. Poisson processes. Along with Brownian motion Poisson process is a basic process in probability theory. The particular topics include: a. Poisson process and rare events b. Conditioning. Connections with empirical process. c. Distribution of the jump moments (arrival times) d. Introduction to queueing systems

7. Renewal processes. The particular topics include: a. Definition and examples b. Renewal equation c. Renewal paradox d. Examples and applications

8. Spacial Poisson processes. The particular topics include: a. Modeling of spacial events - random point process in space, definition and examples b. Independence of increments c. Conditioning and connection with empirical process for vector random variables

### Course Disclaimer

Courses and course hours of instruction are subject to change.

Eligibility for courses may be subject to a placement exam and/or pre-requisites.

Some courses may require additional fees.

Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.

Please reference fall and spring course lists as not all courses are taught during both semesters.

Availability of courses is based on enrollment numbers. All students should seek pre-approval for alternate courses in the event of last minute class cancellations

Please note that some courses with locals have recommended prerequisite courses. It is the student's responsibility to consult any recommended prerequisites prior to enrolling in their course.