What is Random variable, distribution?

Also known as: random variable · distribution

Definition 18.5 High School Mathematics · Chapter 18 — Probability and Random Variables

A random variable on $\Omega$ is a function $X$ assigning a number to each outcome. Its distribution is the table of its possible values $x_1, \dots, x_k$ together with their probabilities

\P(X = x_i) = \P\bigl(\{\omega : X(\omega) = x_i\}\bigr), \qquad \sum_{i=1}^k \P(X = x_i) = 1 .

Examples

Example 18.6

A game: pay $2$ euros, roll one fair die, receive the value shown if it is at least $5$ , nothing otherwise. Let $G$ be the gain (received minus paid). The outcomes $1, 2, 3, 4$ give $G = -2$ ; the outcome $5$ gives $G = 3$ ; the outcome $6$ gives $G = 4$ . The distribution of $G$ :

$g$	$-2$	$3$	$4$
$\P(G = g)$	$\dfrac46$	$\dfrac16$	$\dfrac16$

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Definition 33.1 High School Mathematics · Chapter 33 — Random Variables and the Binomial Distribution

A random variable on a finite sample space $\Omega$ is a function $X \colon \Omega \to \R$ . Its distribution (or law) is the data of its possible values $x_1, \dots, x_k$ and of the probabilities

p_i = \P(X = x_i), \qquad \sum_{i=1}^{k} p_i = 1 .

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Examples

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