# Discrete Random Variable

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A discrete random variable is a random variable associated with a countable sample space.

**Context:**- It must be associated to a Probability Mass Function.
- It must take only a Countably Infinite Number of Distinct Values.
- It can map the Sample Space Ω onto a Countable Set.
- It can be denoted (typically) by italicized uppercase letters, such as [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], or [math]\displaystyle{ Z }[/math].
- by [math]\displaystyle{ X=x_1 }[/math]

**Example(s):**- a Categorical Random Variable, such as a binary random variable.
- an Ordinal Random Variable.
- …

**Counter-Example(s):****See:**Discrete Probability Space, Countable Set, Probability Mass Function.

## References

### 2006

- (Dubnicka, 2006c) ⇒ Suzanne R. Dubnicka. (2006). “Random Variables - STAT 510: Handout 3." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
- MATHEMATICAL DEFINITION: A random variable [math]\displaystyle{ X }[/math] is a function whose domain is the sample space S and whose range is the set of real numbers [math]\displaystyle{ R }[/math] = {
*x*: −∞ < [math]\displaystyle{ x }[/math] < ∞.}. Thus, a random is obtained by assigning a numerical value to each outcome of a particular experiment. - WORKING DEFINITION: A random variable is a variable whose observed value is determined by chance.
- NOTATION: We denote a random variable [math]\displaystyle{ X }[/math] with a capital letter; we denote an observed value of [math]\displaystyle{ X }[/math] as [math]\displaystyle{ x }[/math], a lowercase letter.
- TERMINOLOGY : The support of a random variable [math]\displaystyle{ X }[/math] is set of all possible values that [math]\displaystyle{ X }[/math] can assume. We will often denote the support set as S
_{X}. If the random variable [math]\displaystyle{ X }[/math] has a support set*S*that is either finite or countable, we call [math]\displaystyle{ X }[/math] a 'discrete random variable._{X}

- MATHEMATICAL DEFINITION: A random variable [math]\displaystyle{ X }[/math] is a function whose domain is the sample space S and whose range is the set of real numbers [math]\displaystyle{ R }[/math] = {

### 1986

- (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
- '
*Definition 3.2.1. A real-valued function whose domain is the sample space S is called a*random variable*. We denote random variables by uppercase letters, often [math]\displaystyle{ X }[/math],*Y, or*Z*. - If the range of the mapping contains either a finite or countably infinite number of values, the random variable is said to be
*discrete*; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be*continuous*. - Associated with each discrete random variable [math]\displaystyle{ Y }[/math] is a
*probability density function*(or pdf*). “f*(_{Y}*y*). By definition,*f*(_{Y}*y*) is the sum of all the probabilities associated with outcomes in [math]\displaystyle{ S }[/math] that get mapped into [math]\displaystyle{ y }[/math] by the random variable*Y*. That is.*f*(_{Y}*y*) =*P*({*s*(∈)*S*|*Y*(*s*) =*y*})

- Conceptually,
*f*(_{Y}*y*) describes the probability structure induced on the real line by the random variable Y. - For notational simplicity, we will delete all references to [math]\displaystyle{ s }[/math] and [math]\displaystyle{ S }[/math] and write:
*f*(_{Y}*y*) =*P*(*Y*(*s*)=*y*). In other words,*f*(_{Y}*y*) is the "probability that the random variable Y takes on the value*y*."

- '