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4.1 Definition

A random variable ${R}$ is a function ${R:\Omega{\rightarrow}\mathbb{R}}$ . For each outcome in the sample space, ${\omega\in\Omega}$ , it associates a unique real number ${R(\omega)}$ .

Intriguingly, a random variable is neither random nor a variable!

Example 4.1.

Suppose ${\Omega=\{(i,j):i,j=1,\ldots,6\}}$ is the sample space resulting from rolling two dice. We can define random variables by

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$R((i,j))=i+j$ , the sum of the values on the dice,
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$S((i,j))=\max\{i,j\}$ , the bigger of the two values on the dice.

Every time the experiment is conducted exactly one value of the random variable is observed; this is called a realisation of the random variable.

The range of values taken by the random variable ${R}$ defined on ${\Omega}$ , that is ${\{R(\omega):\omega\in\Omega\}}$ , is known as the induced sample space for ${R}$ and is sometimes written as ${{\mathcal{S}}}$ .

In this chapter we shall focus on discrete random variables — that is functions where ${{\mathcal{S}}}$ is finite or countable e.g. ${{\mathcal{S}}=\{\ldots,-2,-1,0,1,2,\ldots\}}$ . We will move on to continuous random variables — functions where ${{\mathcal{S}}}$ is uncountable e.g. ${{\mathcal{S}}={\mathbb{R}}}$ — in Chapter 6.

Random variables are important as:

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they are the result of most real experiments
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additional structure imposed by the number system, such as ordering, enables further development of the ideas in the previous chapters.

4.1.1 Introductory examples of random variables