**(To be used as a companion to the course in Environmental Data Analysis)**

**Developed by:**

**Roy W. Koch**

**Professor of Civil Engineering and Environmental Sciences**

**Portland State University**

**Introduction**

In this tutorial, we present a brief review of the basic concepts of probability and statistics. Although it is assumed that each student has the appropriate prerequisite for this class, a calculus-based course in probability and statistics, it is also assumed that not everyone uses these concepts regularly and that a review or reference document would be useful. We will use the prerequisite as the starting point for our discussions so it important that you are familiar with the concepts presented in this tutorial. For the most part, we will use the mathematical basis of the ideas of probability and statistics to present probability models and statistical methods as they apply to the analysis and description of environmental data. We will not do mathematical calculations (e.g. integration, etc.) as part of the course work although mathematics will be used to explain various concepts, as appropriate. In the process of reviewing the basic concepts some additional ideas may be presented so a review of the tutorial is suggested for all participants.

**Organization of the tutorial**

The review has a fairly standard organization, beginning with the underlying concepts of random phenomena including random variables, samples spaces and events and proceeds to develop the basic laws of probability based on the nature of these events and the measures of probability associated with them. These are very useful ideas, which underlie all of the subsequent discussion and form the intuitive basis for "thinking" probabilistically. Following these very basic ideas is a definition of probability distributions and related functions including expected values and moments. Then, a brief discussion of a number of particular probability distributions that have been successfully used in describing environmental data is presented. Finally, a review of the basic notions of moments and the estimation of moments and the parameters of probability distributions is presented. In these descriptions, an effort has been made to relate the mathematical concepts to some other common concepts in science and engineering (e.g. the centroid and the mean value) and to provide geometric descriptions where appropriate to allow the reader to achieve a more intuitive understanding of the material.

The material presented is all basic material in probability and statistics and can be found in any number of references. Several are presented in the list of references.

Taken together, the ideas presented in this tutorial provide the basic background required understanding and applying the methods presented in the course. You can move directly to any one of the four basic sections via the listing below.

Probability Concepts

Distributions, expected value and moments

Probability distributions

Moments and parameter estimation

References