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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 4

Distributions

In Chapter 2, we were introduced to random variables. In nature and in finance, random variables tend to follow certain patterns, or distributions. In this chapter we will learn about some of the most widely used probability distributions in risk management.

PARAMETRIC DISTRIBUTIONS

Distributions can be divided into two broad categories: parametric distributions and nonparametric distributions. A parametric distribution can be described by a mathematical function. In the following sections we will explore a number of parametric distributions including the uniform distribution and the normal distribution. A nonparametric distribution cannot be summarized by a mathematical formula. In its simplest form, a nonparametric distribution is just a collection of data. An example of a nonparametric distribution would be a collection of historical returns for a security.

Parametric distributions are often easier to work with, but they force us to make assumptions, which may not be supported by real-world data. Nonparametric distributions can fit the observed data perfectly. The drawback of nonparametric distributions is that they are potentially too specific, which can make it difficult to draw any general conclusions.

UNIFORM DISTRIBUTION

For a continuous random variable, X, recall that the probability of an outcome occurring between b1 and b2 can be found by integrating as follows:

where f(x) is the probability density function (PDF) of X.

The uniform distribution ...

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