Introduction to Bayesian Estimation and Copula Models of Dependence

1Random Variables and Distributions

Chapter 1 is by no means suggested to replace or replicate a standard course in probability. Its purpose is to provide a reference source and remind the readers what topics they might need to review. For a systematic review of probability and introduction to statistics we can recommend excellent texts by DeGroot and Schervish [4], Miller and Miller [10], and Rice [12]. In-depth coverage of probability distributions in the context of loss models is offered by Klugman et al. in [8]. If the reader is interested in a review with a comprehensive software guide, we can recommend Crawley’s handbook in R [3].

Here we will introduce the main concepts and notations used throughout the book. The emphasis is made on the simplicity of explanations, and often in order to avoid technical details we have to sacrifice mathematical rigor and conciseness. We will also introduce a library of distributions for further illustrations. Without a detailed reference to the main facts of probability theory, we need to however emphasize the role played in the sequel by the concept of conditional probability, which becomes our starting point.

1.1 Conditional Probability

Let A and B be two random events, which could be represented as two subsets of the same sample space S including all possible outcomes of a chance experiment: A⊆S and B⊆S. Conditional probability of B given A measures the chances of B to happen if A is already known to occur. It can be defined for events ...