An Introduction to Financial Mathematics, 2nd Edition

Chapter 3

Random Variables

3.1    Introduction

We have seen that outcomes of experiments are frequently real numbers. Such outcomes are called random variables. Before giving a formal definition, we introduce a convenient shorthand notation for describing sets involving real-valued functions X, Y etc. on Ω. The notation essentially leaves out the standard “ω ∈ Ω” part of the description, which is often redundant. Some typical examples are

$\left\{X<a\right\}≔\left\{\omega \in \Omega |X\left(\omega \right)<a\right\}\text{and}\left\{X\le Y\right\}≔\left\{\omega \in \Omega |X\left(\omega \right)\le Y\left(\omega \right)\right\}.$

We also use notation such as {X ∈ A, Y ∈ B} for {X ∈ A} ∩ {Y ∈ B}. For probabilities we write, for example, $\mathbb{P}\left(X<a\right)$ rather than $\mathbb{P}\left(\left\{X<a\right\}\right)$. The following numerical example should illustrate the basic idea.

3.1.1 Example. The table below summarizes the distribution ...