Loss Models, 5th Edition

5Continuous Models

5.1 Introduction

In this chapter, a variety of continuous models are introduced. The collection developed here should be sufficient for most modeling situations. The discussion begins by showing how new distributions can be created from existing ones. This bottom-up approach allows for additional parameters to be added.

5.2 Creating New Distributions

Many continuous distributions are listed in Appendix A where formulas are given for the pdf, cdf, and other quantities. In actuarial applications, we are mainly interested in distributions that have only positive support, that is, where . Distributions with this property that are familiar to most students of statistics include the exponential, gamma, Pareto, and lognormal distributions.

For any distribution, it is possible to construct other, new distributions by making a transformation or by using some other mathematical technique. Many of the distributions in Appendix A can be obtained by applying such methods to other distributions that are also listed in Appendix A. In this section, we illustrate such methods through numerous examples. The examples help explain some of the relationships between the distributions. In particular, Section 5.3 examines “families” of distributions where the members of the family are all special cases of a “parent” distribution.

5.2.1 Multiplication by a Constant

This transformation ...