# Discrete Probability Theory

## Module IN0018

This Module is offered by TUM Department of Informatics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2015 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions | |
---|---|

SS 2015 | WS 2011/2 |

### Basic Information

IN0018 is a semester module in German language at Bachelor’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

- Further Modules from Other Disciplines

Total workload | Contact hours | Credits (ECTS) |
---|---|---|

180 h | 75 h | 6 CP |

### Content, Learning Outcome and Preconditions

#### Content

++ 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

#### Learning Outcome

- 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.

#### Preconditions

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

Type | SWS | Title | Lecturer(s) | Dates |
---|---|---|---|---|

VO | 3 | Discrete Probability Theory (IN0018) | ||

UE | 2 | Discrete Probability Theory, Exercise Session (IN0018) |

#### Learning and Teaching Methods

#### Media

#### Literature

- 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

### Module Exam

#### Description of exams and course work

#### Exam Repetition

The exam may be repeated at the end of the semester.