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).