# Distribution Functions

This chapter covers the main definitions and properties of probability distributions that are relevant for risk management techniques. Examples of risk applications are also introduced when possible. The first section provides generic definitions and terminology, plus conventions used in the text for representing random variables. The inverse functions, presented in Section 10.2 serve repeatedly in this text for simulation purposes. The section also relates to uniform distributions from Section 10.4. Each section is dedicated to distributions that are of interest in risk management. There are numerous publications on statistical distributions, statistical sampling and properties of well-known distributions^{[1]}.

Putting together the distributions simplifies subsequent developments. Readers can refer to this chapter when needed, as long as they have in mind the basic definitions of random variables, of distribution functions, and of moments.

## RANDOM VARIABLES AND PROBABILITY DISTRIBUTION FUNCTIONS

Deterministic variables take a value fully determined by the existing information. Random variables, also called stochastic variables, can take various values associated with probabilities. Probabilities characterize the chance that a random variable falls within a preset range of values.

Random variables are conditional on an information set that evolves with time. For example, the stock price as of date *t* is conditioned by the set of information available as of *t.* When *t* increases, the information set increases and the stock price adjusts accordingly to good and bad news. Random variables are designated by capital letters and their particular values are designated by small letters. Hence *X* is a random variable which can take a value x.

A random variable takes values associated with a probability. The probability measures the "chances" of a random variable taking any particular value or falling within a range of values. The probability that a random variable falls within the allowed range of possible values is always 1. The range of permissible values is the "support" of the distribution. Hence all probabilities assigned to all permissible values sum up to 1.

- [1] Basic properties are shown in reference 78 (for example), which includes sampling errors distributions. Very detailed presentations of all univariate continuous distributions are in references 41 and 42, which can serve as reference for most distributions and the related methodologies to deal with most statistical issues.