# Discrete Random Variable

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

## 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 $\displaystyle{ X }$ is a function whose domain is the sample space S and whose range is the set of real numbers $\displaystyle{ R }$ = {x : −∞ < $\displaystyle{ x }$ < ∞.}. 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 $\displaystyle{ X }$ with a capital letter; we denote an observed value of $\displaystyle{ X }$ as $\displaystyle{ x }$, a lowercase letter.
• TERMINOLOGY : The support of a random variable $\displaystyle{ X }$ is set of all possible values that $\displaystyle{ X }$ can assume. We will often denote the support set as SX. If the random variable $\displaystyle{ X }$ has a support set SX that is either finite or countable, we call $\displaystyle{ X }$ a 'discrete random variable.

### 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 $\displaystyle{ X }$, 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 $\displaystyle{ Y }$ is a probability density function (or pdf). “fY(y). By definition, fY(y) is the sum of all the probabilities associated with outcomes in $\displaystyle{ S }$ that get mapped into $\displaystyle{ y }$ by the random variable Y. That is.
• fY(y) = P({s(∈)S |Y(s) = y})
• Conceptually, fY(y) describes the probability structure induced on the real line by the random variable Y.
• For notational simplicity, we will delete all references to $\displaystyle{ s }$ and $\displaystyle{ S }$ and write: fY(y) = P(Y(s)=y). In other words, fY(y) is the "probability that the random variable Y takes on the value y."