MATHEMATICAL STATISTICS, BASIC COURSE | FMS140 |

**Aim**

The course is intended to give the student the basics in mathematical modelling of random variation and an understanding of the principles behind statistical analysis. It shall also give the students a toolbox containing the most commonly used models and methods, as well as the ability to use these in practical situations.

The course fills two purposes, providing a fundamental knowledge of mathematical statistics, as well as giving a foundation for further studies.

The fundamental knowledge is essential for those who, in their professional lives, will not necessarily be involved in statistical analyses on a daily basis, but who, on occasion, will be expected to perform basic statistical tests and present the resultsĀ to their colleagues. They will also be expected to be able to read and assess the analyses of others.

The course shall also give a basis for further studies, particularly in design of experiments and risk evaluation.

*Knowledge and understanding*

For a passing grade the student must

- be able to relate environmental statistical questions regarding random variation and observed data to the concepts of random variables, distributions, relationships between variables, and dependent data,
- be able to explain the concepts of independence, probability, distribution, expectation, and variance,
- be able to calculate the probability of an event and the expectation from a given distribution,
- be able to describe fundamental techniques for statistical inference and be able to use them on basic statistical models.

*Skills and abilities*

For a passing grade the student must

- be able to construct a simple statistical model describing a problem based on a real life situation or on a collected data material,
- be able to examine a statistical model and its ability to describe reality,
- be able to use a computational program for simulationĀ and interpretation of statistical models, as well as for data analysis,
- be able to choose, perform, and interpret a statistical procedure that answers a given statistical problem,
- be able to use statistical terms within the field both written and orally,
- be able to present a statistical analysis in a technical report,
- examine a statistical analysis of a data material and present the judgement orally.

**Contents**

The course contains fundamental concepts in probability theory, inference theory, regression analysis, and time series analysis.

In probability theory the concepts used are random variables and distributions for describing variation and random phenomena, often related to applications in environmental statistics. Different distributions, such as binomial, Poisson, normal, exponential, and log normal distributions, are studied and the concept of expectation and variance of a distribution is introduced. Special attention is paid to the normal distribution and its property as a limit distribution. Simulations from the distributions and studies of the models are performed in Matlab. This part constitutes approximately 2/7 of the course.

In inference theory we start with observed data and estimate parameters in simple probability models, and describe the uncertainty of the estimates. Emphasis is placed on the relationship between the model and the reality based problem, as well as the conclusions that can be drawn from observed data. In this analysis we use basic techniques, such as confidence intervals and hypothesis testing. This part constitutes approximately 2/7 of the course.

In regression analysis we study how the relationship between two or more variables can be described. Most often the relationship will be linear. Often in environmental applications one of the variables is a time variable which leads to trend analysis. We study different techniques for comparison and choice between different models for relationships. Environmental data if often dependent and therefore we introduce time series with concepts of trend, season, and noise. Techniques, such as auto-correlation function, are used to describe the dependence. A simple AR(1) model for dependent data is introduced. This part, resting heavily on the use of Matlab, constitutes approximately 3/7 of the course.

**Literature**

Olbjer, L: Experimentell och industriell statistik. Lund 2000.