*2*Random Variables

## 2.1 Introduction

An actuarial model is a representation of an uncertain stream of future payments. The uncertainty may be with respect to any or all of occurrence (is there a payment?), timing (when is the payment made?), and severity (how much is paid?). Because the most useful means of representing uncertainty is through probability, we concentrate on probability models. For now, the relevant probability distributions are assumed to be known. The determination of appropriate distributions is covered in Chapters 10 through 15. In this part, the following aspects of actuarial probability models are covered:

- Definition of random variable and important functions, with some examples.
- Basic calculations from probability models.
- Specific probability distributions and their properties.
- More advanced calculations using severity models.
- Models incorporating the possibility of a random number of payments, each of random amount.

The commonality we seek here is that all models for random phenomena have similar elements. For each, there is a set of possible outcomes. The particular outcome that occurs will determine the success of our enterprise. Attaching probabilities to the various outcomes allows us to quantify our expectations and the risk of not meeting them. In this spirit, the underlying random variable will almost always be denoted with uppercase italic letters near the end of the alphabet, such as *X* or *Y*. The context will provide a name and some likely characteristics. ...

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