autumn 2018

## STA-2001 Stochastic Processes - 10 ECTS

## Admission requirements

Applicants from Nordic countries: Generell studiekompetanse og følgende spesielle opptakskrav:

Matematikk R1 + R2 og i tillegg enten:

- Fysikk 1 + 2 eller
- Kjemi 1+ 2 eller
- Biologi 1 + 2 eller
- Informasjonsteknologi 1 +2 eller
- Geofag 1 + 2 eller
- Teknologi og forskningslære 1 + 2

International applicants: Higher Education Entrance Qualification and certified language requirements in English. It is a requirement that students have some prior knowledge of biology and ecology, chemistry and mathematics (Participants must have taken introductory level university courses, and achieved pass grades, in these subjects).

A list of the requirements for the Higher Education Entrance Qualification in Norway can be found on the Norwegian Agency for Quality Assurance in Education website - nokut.no

Application code: 9336 (Nordic applicants).

## Course content

This course builds on STA-1001 Probability and statistics. The course is a continuation of the probability theory presented in STA-1001 with an emphasis on construction, interpretation and analysis of probability models for simple processes or dynamic systems. Discusses conditional probability, conditional expectations, Markov chains, Poisson processes, branching processes, birth and death processes and other stochastic processes.## Objectives of the course

This course gives students an introduction to applied probability theory and stochastic processes, including use of conditioning as an important tool for probability computations. Within stochastic processes, primary emphasis is placed upon the analysis of models with countable state space in discrete or continuous time. Of special importance is that students have a command of different types of Markov processes, including Poisson processes and birth and death processes.

The student shall:

- be able to use basic probability theory. Here it is important to be able to do computations with stochastic variables, with one- or multi-dimensional distributions. Special importance is attached to conditional probability and conditional expectation, and to be able to use these as tools in probability computations and stochastic models.
- be able to set up and analyze Markov models in discrete time. Here it is important to be able to express Markov models by means of transition matrices and to compute the probability for transitions in one or more steps. One must be able to classify states, find expected time in states and limit probabilities for different states. One should also be able to identify and utilize the fact that a process is time-reversible and to be able to analyze the special case of branching processes.
- have fundamental knowledge of Poisson processes. Here it is important to understand the distribution in time between occurrences, between a given number of occurrences, and conditional distribution of occurrence times. In conjunction with this, the exponential distribution and its properties are important. One should have knowledge of extensions of the Poisson model: non-homogeneous, conditional, and compound Poisson processes.
- be able to set up and analyze Markov models in continuous time. Here it is important to be able to express models with the help of transition rates, and to find the probability for transition with the help of differential equations. One should also be able to find the limiting probabilities given by balance equations, and be able to recognize and utilize that a process is time-reversible. Special emphasis is given to birth and death processes, including the expected number of individuals, expected time to reach a certain number of individuals, transition probabilities and limiting distributions for these.

## Assessment

Written final exam of 4 hours duration, counting 100 %.

Assessment scale: Letter grades A-F.

Postponed examination: Students with valid grounds for absence will be offered a postponed examination early in the following semester.

For further details see:

- Utfyllende bestemmelser for eksamener ved Fakultet for naturvitenskap og teknologi (only in Norwegian)

- Regulations for examinations at the University of Tromsø

**Coursework requirements **

A passing grade is required on the mandatory homework sets for permission to take the exam.

## Date for examination

Written test 29.11.2018The date for the exam can be changed. The final date will be announced in the StudentWeb early in May and early in November.

## Recommended reading/syllabus

**Syllabus for STA-2001 Stochastic Processes, autumn 2018**

UiT The Arctic University of Norway, Department of Mathematics and Statistics

**Tetbook: **Sheldon M. Ross, "Introduction to Probability Models". Academic Press, 10 th. edition

Chapter 1 Introduction to Probability Theory

Chapter 2 Random Variables

Chapter 3. 1 - 3.6 Conditional Probability and Conditional Expectation

Chapter 4.1 - 4.9 Markov Chains

Chapter 5. 1 - 5.4 The Exponential Distribution an the Poisson Process

Chapter 6.1 - 6.6 Continuous-Time Markov Chains

Chapter 11.1 - 11.2, 11.5 Simulation