random variable

Also: stochastic variable

Informally, a random variable is a quantity whose value is a real number that depends in some way on chance. Examples:

The number of eyes thrown with a die
Taking a sample from a population
The height of a person
...

Formally, a random variable is defined as a function that assigns a real number to each possible outcome of an experiment. The set of all possible outcomes of an experiment is called the universe or outcome space, denoted as \(\Omega\) (Greek capital letter Omega).

\[X: \Omega \to \mathbb{R}: \omega \mapsto X(\omega)\]

Example: the total number of eyes thrown with two dice can be formalized as follows:

\[X: \Omega \to \mathbb{R}: (a,b) \mapsto a+b\]

With \((a, b) \in \Omega = \{ (1,1), (1,2), \ldots, (6,6) \}\) (= all possible combinations of throwing 2 dice). The value range of \(X\) is then \(\{2, 3, \ldots, 12\}\).

If the value range of a random variable \(X\) is finite (as in this example) or countably infinite (e.g. \(\mathbb{N}, \mathbb {Z}, \mathbb{Q}\)), we speak of a discrete random variable. If the range of values is uncountably infinite (e.g. \(\mathbb{R}, \mathbb{R^+}\)), we speak of a continuous random variable.

For a random variable \(X\), we are particularly interested in the probability distribution of \(X\).