Discrete Random Variable

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):
  • a Continuous Random Variable.
• See: Discrete Probability Space, Countable Set, Probability Mass Function.

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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_X that is either finite or countable, we call [math]\displaystyle{ X }[/math] 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 [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."

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