Loss Data Analysis

3Individual severity models

In this chapter we shall examine a few of the large variety of possible continuous, positive random variables that are used to model individual loss severities. Besides the obvious listing of their densities and statistical properties, of potential interest for us is the fact that the Laplace transform of the probability density (in short, the Laplace transform of the random variable) can be explicitly computed. Let us begin with the basic definition.

Definition 3.1. Let X be a positive random variable with density fX(x). Its Laplace transform ψ(α) is defined for any α > 0 by

$\varphi \left(\alpha \right)=\text{E}\left[e^{-\alpha X}\right]={\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\alpha x}{f}_X\left(x\right)dx.}$

A widely used and related concept is the moment generating function, given by M(t) = ϕ(−t). Those who get annoyed ...