Random Variables

Chapter 4. Probability Distributions: Random Variables

Random Variables

Random Variable
A random variable is a variable that assigns a numerical value to each outcome in the sample space of a random experiment.

Random variables are usually denoted by capital letters from the Roman alphabet (e.g. $X$ , $Y$ ).

A random variable can either be discrete or continuous.

Range of Random Variable
The set of all possible values that a random variable $X$ can take on is called the range of the variable and is denoted by $R(X)$ .

In this course, we will use the American notation for intervals when describing the range of continuous random variables:

An open interval, which does not include the endpoints, is symbolized by round brackets.
- $(a,b)$
A closed interval, which does include the endpoints, is symbolized by square brackets.
- $[a,b]$
A half-open interval, which includes one endpoint but not the other, is symbolized by a combination of round and square brackets.
- $[a,b)$
- $(a,b]$

Consider the random experiment of rolling two dice with six sides, numbered from $1$ to $6$ , and observing the numbers on top.

Let $X$ be the sum of the upward-facing numbers. In this case, $X$ can take on any integer between $2$ and $12$ , inclusive.

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

$X$ is a discrete random variable.

Let $T$ be the time in hours until a light bulb fails after it is first illuminated.
$R(T)=[0, \infty)$

$T$ is a continuous random variable.

Let $Y$ be the number of planes waiting to land at Schiphol airport.
$R(Y)=\{0,1,2, \ldots\}$

$Y$ is a discrete random variable.