NMAK19003U Applied Probability

Volume 2024/2025

MSc Programme in Actuarial Mathematics 
MSc Programme in Statistics
MSc Programme in Mathematics-Economics


Applied Probability is an area which develops techniques for the use in stochastic modelling.

In this course we use concepts and tools from Markov processes, renewal theory, random walks and (optionally) themes like Markov additive processes and regeneration.

The class of phase-type distributions, defined in terms of absorption times in Markov processes, will play a major role througout the course. They constitute a class of distributions which may approximate any positive distribution arbitrarily close, and they provide for elegant solutions to complex problems by using probabilistic arguments often relying on sample path arguments and leading to explicit formulae expressed in terms of matrices. 

Their interplay with ladder height methods (in random walks), provide important applications e.g. ruin theory in  non-life insurance, where also Markov additive processes (Markov modulation) may be used, and in queueing theory regarding waiting time distributions. 

Though phase-type distributions are light-tailed, appropriate transformations give rise to dense classes of heavy-tailed distributions with e.g. Pareto, Weibull or Mittag-Leffler type of tails, in which basic distributional properties can be expressed in terms of functions of matrices again. These can be used in a similar way as phase-type distributions, and will be employed in the modelling of extremal events (e.g. insurance claims). 

The fitting of phase-type and/or their heavy-tailed counterpart to data will also be considered as this constitues an important part of their applicability.

The course is self-contained, providing all necessary background from both theory, applications and estimation. Students, who come around topics they have encountered previously, will benefit from reviewing the material in this alternative and highly probabilistic context, enabling a deeper understanding of the underlying subjects.

Learning Outcome

At the end of the course the student is expected to have:

  • The ability to employ the classical tools from Applied Probability for solving stochastic models by performing probabilistic (sample path) arguments.
  • Knowledge about renewal theory, random walks, Markov processes, phase-type distributions, ladder height distributions, ruin probabilities, severity of ruin, heavy-tailed modelling of extremal events.
  • Skills to formalize phase-type distributions and their transformed counterparts, discuss their theoretical background, and apply them in the modelling of risk and extremal events. 
  • Competences to idenitify patterns of random phenomena and building adequate stochastic models which can be solved for by using Markov processes and related techniques.

M. Bladt & B. F. Nielsen (2017) Matrix-exponential distributions in Applied Probability. Springer Verlag

Probability theory at bachelors level, including measure theory. Some previous exposure to stochastic processes will be an advantage.

Academic qualifications equivalent to a BSc degree is recommended.
7 weeks of lectures (2 x 2 hours per week) combined with theoretical and practical exercises (2 hours per week).
  • Category
  • Hours
  • Lectures
  • 28
  • Preparation
  • 163
  • Exercises
  • 14
  • Exam
  • 1
  • Total
  • 206
Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Oral examination, 30 minutes (30-minute preparation time)
All aids allowed

During preparation only.

Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners

Same as the ordinary exam

Criteria for exam assesment

The student must in a satisfactory way demonstrate that they have mastered the learning outcome of the course.