Random Variables

4. Probability Distributions: Random Variables

Random Variables

Random Variable

A random variable is a variable for which you cannot know its outcome before it is observed. The set of values it may take are however known a priori. The formal definition for a random variable is: a variable that maps each outcome in the sample space of a random experiment to a numerical value.

Random variables are usually denoted by capital letters from the end of the Roman alphabet (e.g. $X$ , $Y$ ). They can have any measurement level and, when numerical, be discrete or continuous.

Range of Random Variable

The set of values which a random variable $X$ can take is called the range of the random variable and is denoted by $R(X)$ .

Consider the random experiment of rolling two dice. Let $X$ be the sum of the upward-facing dots. In this case, $X$ can take on any value between $2$ and $12$ .

$R(X) = \{2,3,\ldots,12\}$

$X$ is discrete.

Let $T$ be the time until a light bulb fails after it is first illuminated.

$R(T)=[0, \infty)$

$T$ is continuous.

Let $Y$ be the number of planes waiting to land at Schiphol airport.

$R(Y)=\{0,1,2, \ldots\}$

$Y$ is discrete.

In the examples above, different types of brackets are used. These are not applied randomly, but are used to describe properties of a set.

Braces, $\{$ and $\}$ , are used for discrete random variables. They contain a finite number of discrete values or symbols (which do not all have to be shown).
Square brackets, $[$ and $]$ , and parentheses, $($ and $)$ , are used for continuous random variables. The value following $[$ or $($ is the lower range of the set. This value is part of the set if $[$ is used and not part of the set if $($ is used. The same applies to the upper range. A set without an upper or lower bound is described with $(-\infty$ and $\infty)$ respectively , $[0, \infty)$ describes a set with values $\ge 0$ (ranging from zero to infinity).