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.