Discrete Probability Theory

-Fundamentals of (discrete) probability theory
++ probability spaces, events, principle of inclusion/exclusion, Boolean inequality, conditional probability, multiplication theorem, total probability theorem, Bayesian theorem, independence
++ random variables, expected value, variance, linearity of expected value, conditional random variables and their expected value, variance, moments and central moments, several random variables and their common density & distribution, independence from random variables, moments of composite random variables, indicator variables
++ discrete distributions: Bernoulli distribution, binomial distribution, geometric distribution, coupon collector problem, Poisson distribution, relationships among distributions
++ methods for estimating probabilities, inequalities of Markov and Chebyshev, Chernoff barriers
++ Law of large numbers
++ Probability-generating functions and their application to distributions, moment-generating functions with various applications

-continuous probability spaces
++ continuous random variables, Kolmogorov axioms, sigma algebras, Lebesgue integrals, computing with continuous random variables, simulation of random variables
++ continuous distributions: Equal distribution, normal distribution and linear transformation, exponential distribution and waiting processes, relationship with discrete distributions
++ several continuous random variables, boundary distributions and independence, sums of random variables
++ Moment generating functions for continuous random variables
++ Central limit theorem

-Inductive Statistics
++ estimation variables, maximum likelihood principle, confidence intervals, hypothesis testing, statistical test development and application

-Stochastic processes
++ processes with discrete time, Markov chains, transition probabilities, arrival probabilities, transition times, return times, fundamental theorem for ergodic Markov chains

After successful completion of the module
- Participants are familiar with important concepts of discrete and continuous probability spaces and stochastic processes and can in large part deduce them themselves
- master calculation rules for the determination and estimation of probabilities, expected values and variances,
- are able to map real problems to abstract probability spaces and
- can easily apply simple statistical tests.

IN0015 Discrete Structures, MA0901 Linear Algebra for Informatics, MA0902 Analysis for Informatics

The module consists of lectures and tutorials. During the lectures students are asked to solve small exercises online. Students also receive weekly assignments, whose solution is discussed in the tutorials.

Slide show, blackboard, written assignments.

- T. Schickinger, A. Steger: Diskrete Strukturen - Band 2, Springer Verlag, 2001
- Nobert Henze: Stochastik für Einsteiger, Vieweg, 2004
- R. Mathar, D. Pfeifer: Stochastik für Informatiker, B.G. Teubner Stuttgart, 1990
- M. Greiner, G. Tinhofer: Stochastik für Studienanfänger der Informatik, Carl Hanser Verlag, 1996
- H. Gordon: Discrete Probability, Springer-Verlag, 1997
- R. Motwani, P. Raghavan: Randomized Algorithms, Cambridge University Press, 1995
- L. Fahrmeir, R. Künstler, I. Pigeot, G. Tutz: Statistik - Der Weg zur Datenanalyse, Springer-Verlag, 1997

Students will be assessed by a 120 minutes written exam, which consists of three types of exercises. Comprehension exercises test if the student understands the basic concepts and theorems of the lecture. They require students to apply these concepts to examples. Algorithmic exercises test if the student knows and is able to apply the presented rules to selected inputs. Modelling exercises test the ability of the student to use the mathematical tools from the lectures to model and solve concrete problems.