### 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,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]#

- #[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.

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