## Runs in random sequences

Goal:

Imagine you are observing a random binary sequence, i.e. at each step a (biased) coin is flipped, yielding a zero or one with some given probabilities p and 1-p. This is a good model for many repetitive processes, for example the production of some product in a factory (product with/without a defect), the failure of a hard drive (bit is saved correctly/incorrectly) etc. A run of length k is the occurence of k ones in a row. In this project, your task is to verify and visualize several central limit theorems and asymptotics regarding such runs by simulation. Among other things, you will find answers to the following questions: After n steps, how many runs of length k can be observed on average? How likely is it that no run of length k has occured?

Schedule: To be determined with the student(s).

Supervisors: Simon Campese

Difficulty level: Introductory/intermediate (some basic knowledge of probability theory is needed)

Tools: Programming can be done in C/C++, Python, R, Matlab/Octave, Wolfram Mathematica, Maple etc., depending on the knowledge of the student.

Results: To be completed at the end of the project. 