# Probability theory

## Overall Course Objectives

To provide the participants with an intuitive and rigorous understanding of probabilistic concepts. The course will familiarize the student with different sources of variability that occur in technical fields and science. This is partly done by exposure to a number of basic models of widespread application. Introduction of various standard techniques enables the students to perform calculations of probabilistic nature. The course provides the mathematical background for probabilistic methods used in statistics courses as well as courses within technical areas with a significant component of random phenomenona. Examples are models of reliability, formation of queues, traffic, analysis of signals, DNA-sequence modeling, etc.

**Learning Objectives**

**Learning Objectives**

- Apply simple approximate formulas for calculation of probabilities
- Apply operations with random variables to obtain new probability distributions.
- Formulate simple probability models from a verbal description.
- Apply basic definitions and axioms in problem solving (simple problems).
- Apply the concept of conditional probability in the basic formulas and definitions of it.
- Apply the concepts of conditional distribution and conditional moments.
- Choose the correct probabilistic model for a real world phenomenon based on the characteristics.
- Perform simple calculations with moments, cross-moments and correlations.
- Apply and utilize the relationship between the different ways of expressing a probability distribution.
- Calculate distributions and quantities derived from the bivariate normal distribution.
- Correctly apply characteristics of discrete and continuous distributions respectively

**Course Content**

**Course Content**

Axioms of probability theory, exclusion- inclusion, conditional probability, independence, Bayes rule, calculation of the probability of a sequence of events, the binomial distribution, normal approximation to the binomial distribution, sampling with and without replacement, the hypergeometric distribution, discrete random variables, moments with emphasis on mean value and variance, Markovs and Chebychevs inequalities, the central limit theorem, indicator variables, the geometric and negative binomial distributions, the Poisson distribution, continuous random variables, the normal- exponential- and gamma- distributions, survival function, hazardrate, change of variable principle in one dimension, cumulative distribution function, order statistics, uniform continuous random variables, bivariate continuous random variables, the Rayleigh distribution, the chi square distribution, conditional distributions, conditional moments, covariance, the bivariate normal distibution

**Teaching Method**

**Teaching Method**

Lectures, exercises and homework problems. The student is required to hand in 3 sets of home work. The home work needs to be approved for the student to register for the exam.

**Faculty**

**Faculty**

## Bo Friis Nielsen

**Remarks**

**Remarks**

The course is a general methodological course aimed at all engineering students.The course requires a certain mathematical maturity, e.g. mastering of basic concepts from calculus. The mathematical maturity can be obtained, while following the course at the expense of an additional workload.