Random Variable

probability-theory

Definition

Random Variable

A random variable is a variable who value is a numerical outcome of a random phenomenon. It is typically denoted by a capital letter, like $X$ and is a function that maps outcomes from the sample space $\Omega$ to the set of real numbers $\mathbb{R}$:

$$X:\Omega \to \mathbb{R}$$

where every elementary event $\omega \in \Omega$ gets a real number $X(\omega )=x\in \mathbb{R}$ assigned, i.e. $X$ is a total function. Then:

• The number $x\in \mathbb{R}$ is one realisation of $X$.
• The set $\{X(\omega )\mid \omega \in \Omega \}$ of all values of $X$ is called the feature space of $X$.
• Events are given by pre-images $X^{-1}(\mathcal{X})=\{\omega \in \Omega \mid X(\omega )\in \mathcal{X}\}$ for $\mathcal{X}\subseteq \mathbb{R}$.

Notation

We may use a more descriptive notation, for example:

$$\begin{array}{rl}[X=x]& =X^{-1}(\{x\})\\ [X<0]& =X^{-1}((-\infty ,0))\\ [X\ge 0]& =X^{-1}([-1,1])\end{array}$$

Realisation

Definition

Random Variable Realisation

The image $x=X(\omega )$ of a random variable $X$ is called one realisation of $X$.

Link to original

Intuition

An elementary event $\omega \in \Omega$ of a sample space is one potential outcome of a random experiment.

Example: Rolling two dices could have the elementary event

$${\omega }_0=(\text{die}1\text{shows}2,\text{die}2\text{shows}5)=(2,5)$$

A random variable $X$ is the rule or function that one can apply to this raw outcome to get a number $x\in \mathbb{R}$, the realisation. In this case, the rule is “take the maximum value of two dice”. So, for any elementary event $\omega \in \Omega$, the random variable is:

$$x=X(\omega )=\max \omega$$

The realisation $x$ is the actual number we get after applying the rule $X$ to the specific elementary event $\omega$. For $w_0=(2,5)$, the realisation is:

$$x_0=X((2,5))=\max (2,5)=5$$

Why is a realisation needed?

• A random variable transform as possibly non-computable object $\omega \in \Omega$ into a real number $X(\omega )\in \mathbb{R}$, allowing for computations.
• A random variable allows to distinguish between the outcome of a random experiment and it’s interpretations. For example, $X((2,5))=5=X((5,2))$ leads to the same realisation.