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An Introduction to Probability Theory and Stochastic Processes with Applications to Biology
Last Updated: 2026-02-05 15:35:50
Abstract
Biology is becoming increasingly quantitative and mathematical modeling is now an integral part of biological research. In many biological processes, ranging from gene-expression to evolution, randomness plays an important role that can only be understood using stochastic models. This course will provide the students with a theoretical foundation for developing such stochastic models and analyzing
Objective
The aim of this course is to introduce certain topics in Probability Theory and Stochastic Processes that have been specifically selected with an eye on biological applications. This course will teach students the tools and techniques for modeling and analyzing random phenomena. Throughout the course, several biological applications will be discussed and students will be encouraged to do additional reading based on their research interests.
Content
The first half of the course will cover the basics of Probability Theory while the second half will delve into the theory of Stochastic Processes. Below is the list of topics that will be covered in the course. 1. The mathematical representation of random phenomena: The probability space, properties of the probability measure, Independence of events, Conditional probability and Bayes formula, applications to parameter inference. 2. Random Variables and their distributions: Discrete and continuous random variables, Expectation and Variance, Important Examples of Random Variables, Independent random variables and their sums, Conditional Distribution and Conditional Expectation, Markov and Chebyshev inequalities. Law of total variation, estimation of intrinsic and extrinsic noise in biological systems. 3. Convergence of Random Variables: Modes of convergence, Laws of large numbers, the central limit theorem, the law of the iterated logarithm, Applications to the analysis of cell population data. 4. Generating functions and their applications: Definition and important examples, Random Walks, Branching processes, Coalescent processes, Modeling epidemic processes and stem-cell differentiation. 5. Markov chains: Transition functions and related computations, Classification of states and classification of chains. Concepts of recurrence, transience, irreducibility and periodicity, Stationary distributions, Continuous time Markov Chain model of a biochemical reaction network. 6. Stochastic Processes: Existence and Construction, Stationary Processes, Renewal Processes, The Wiener Process, The Ergodic Theorem, Leveraging experimental techniques in Biology. 7. Introduction to the theory of Martingales: Basic definitions, Martingale differences and Hoeffding's inequality, Martingale Convergence Theorem, Crossings and convergence, Stopping times and the optional sampling theorem, Doob's maximal inequalities, Applications to the analysis of stochastic biochemical reaction networks.
Resources
Literature
While no specific textbook will be followed, much of the material and homework problems will be taken from the following books: An Introduction to Stochastic Processes with Applications to Biology, Linda Allen, Second Edition, Chapman and Hall, 2010. Probability And Random Processes, Grimmett and Stirzaker, Third Edition, Oxford University Press, 2001.
General Information
- Language
- English
- Levels
- MSC
- Frequency
- Every two years
Examination
- Type
- session examination
- Mode
- written 180 minutes
- Aids
- None
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture with exercise |
An Introduction to Probability Theory and Stochastic Processes with Applications to Biology
Does not take place this semester.
|
No time listed | 3 h weekly |
Offered In
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Computational Biology and Bioinformatics Master (More informations at: )
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Advanced Courses (A total of 30 ECTS needs to be acquired in the Advanced Courses category. Thereof 18 ECTS in the Theory and 12 ECTS in the Biology category. Note that some of the lectures are being recorded: )
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Theory (At least 18 ECTS need to be acquired in this category.)
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Electives (The electives list in the ETH course catalogue is an open list, and the courses listed in the ETH course catalogue provide just examples for possible elective courses, e.g. a selection of eligible courses. Students are expected to look for relevant courses in the ETH and University of Basel course catalogue and ask their mentor for approval. Courses from the advanced course category may also be taken as electives. We particularly recommend browsing the University of Basel course catalogue for elective courses of relevant master's degree programes (using the filter "programe structure" on the course catalogue website), such as for example: Biomedical Engineering, Chemistry, Drug Sciences, Epidemiology, Infection Biology, Molecular Biology, Nanosciences)
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