Random Variable

Definition

Let

• $\mathcal{C}$ : Sample space
• $X$ : Function

If $X$ assigns each element $c\in \mathcal{C}$ one and only one number $X(c)=x$

Then we say $X$ is a random variable

Remark

We denote the space/range of $X$ as $\mathcal{D}=\{x:x=X(c),c\in \mathcal{C}\}$. So,

$$X:\mathcal{C}\to \mathcal{D}$$

Notice that $X(c)\in \mathbb{R}$. Thus unlike $\mathcal{C}$ we have $\mathcal{D}\subseteq \mathbb{R}$

• See also: Example: Random variable representing sum of 2 dice rolls