Loss Models: From Data to Decisions, 4th Edition

A.1 Introduction

Descriptions of the models are given starting in Section A.2. First, a few mathematical preliminaries are presented that indicate how the various quantities can be computed.

The incomplete gamma function¹ is given by

with

A useful fact is Γ(α) = (α − 1) Γ(α − 1). Also, define

At times we will need this integral for nonpositive values of α. Integration by parts produces the relationship

This process can be repeated until the first argument of G is α + k, a positive number. Then it can be evaluated from

However, if α is a negative integer or zero, the value of G(0; x) is needed. It is

which is called the exponential integral. A series expansion for this integral is

When α is a positive integer, the incomplete gamma function can be evaluated exactly as given in the following ...