Random Variable

A Random Variable is a Measurable Real-Valued Function that simply assigns a Number to each Outcome (ω) in a Sample Space (Ω).

• AKA: Random Variable Function.
• Context:
  • Function Input:
    • a Random Experiment Outcome Identifier.
    • Function Domain: a Random Variable Support/Sample Space.
  • Function Output: a Real Number (−∞ < x < ∞).
  • It is a mapping from a Probability Space to a Measurable Space.
  • It can be:
    • a Discrete Random Variable, with a Probability Mass Function.
      • a Categorical Random Variable.
      • a Nominal Random Variable.
    • an Continuous Random Variable, with a Probability Density Function.
    • a Random Variable Vector.
  • Its Output can be the Input to:
    • a Probability Function.
    • an Expected Value Function.
  • It can be denoted:
    • by italicized uppercase letters, such as: X, Y, or Z.
    • without the outcome variable e.g. X(ω) => X and X(ω=Heads) => X(Heads).
• Example(s):
  • A Binary Random Variable, e.g. of a Coin Toss Experiment.
    • X(n)=1 if coin n=H, X(n)=0 if coin n=T.
  • A Random Variable that represents the Sample Space of the Sum of a Two Dice Roll Experiment.
    • X(1,1)=>2, X(5,2)=>7, X(3,4)=>7, ..., X(6,6)=>12
  • A Continuous Random Variable, e.g. of a Lightbulb Lifetime Experiment.
    • X(0)=>0, X(1.101...)=>1.101..., X(268.16...)=>268.16..., ..., X(9362.16...)=>9362.16...
  • A Random Variable that represents a Scalar Statistic (such as Mean Function).
    • X(?ω?)=>??
  • A Vector Random Variable, e.g. ?
    • ??
• See: Probability Space, Correlation Coefficient, Random Experiment.

References

• WordNet
  • a variable quantity that is random.
• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Random_variable
  • In mathematics, random variables are used in the study of chance and probability. They were developed to assist in the analysis of games of chance, stochastic events, and the results of scientific experiments by capturing only the mathematical properties necessary to answer probabilistic questions. Further formalizations have firmly grounded the entity in the theoretical domains of mathematics by making use of measure theory.
  • Intuitively, a random variable is thought of as a function mapping the sample space of a random process to the real numbers.
  • Broadly, there are two types of random variables — discrete and continuous. Discrete random variables take on one of a set of specific values, each with some probability greater than zero. Continuous random variables can be realized with any of a range of values (e.g., a real number between zero and one), and so there are several ranges (e.g. 0 to one half) that have a probability greater than zero of occurring.
  • A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable).
  • Formal definition: Let (Ω, Ƒ, P) be a probability space and (Y, Σ) be a measurable space. Then a random variable X is formally defined as a measurable function X: Omega -> Y. An interpretation of this is that the preimage of the "well-behaved" subsets of Y (the elements of Σ) are events (elements of F), and hence are assigned a probability by P.
• http://en.wiktionary.org/wiki/random_variable
  • (broadly) A quantity whose values are random and to which a probability distribution is assigned, such as the possible outcomes of a roll of a ...

2008

• (Qian) => Gang Qian. (2008). Basic Probability Theory. Lecture Notes: AME 598 Sensor Fusion, Arizona State University, Fall 2008.
  • A random variable X is a function that assigns a real number, X(ζ), to each outcome ζ in the sample space of a random experiment.
  • Let S_X be the set of the sample space.
  • X(ζ) can be considered as a new random experiment with outcomes X(ζ) as a function of ζ, the outcome of the original experiment.

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 X is a function whose domain is the sample space S and whose range is the set of real numbers R = {x : −∞ < 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 X with a capital letter; we denote an observed value of X as x, a lowercase letter.
  • TERMINOLOGY : The support of a random variable X is set of all possible values that X can assume. We will often denote the support set as S_X. If the random variable X has a support set S_X that is either finite or countable, we call X a discrete random variable.
  • The pmf of a discrete random variable and the pdf of a continuous random variable provides complete information about the probabilistic properties of a random variable. However, it is sometimes useful to employ summary measures. The most basic summary measure is the expectation or mean of a random variable X, denoted E(X), which can be thought of as an “average” value of a random variable.
  • TERMINOLOGY : Let X be a discrete random variable with pmf pX(x) and support S_X. The expected value of X is given by E(X) = X x∈SX xpX(x).

1988

• (Dolciani, 1988)
  • Random Variable: "A function that assigns a numerical value to each outcome of an experiment".

1986

• (Larsen & Marx, 1986) => Richard J. Larsen, and Morris L. Marx. (1986). "An Introduction to Mathematical Statistics and Its Applications, 2nd edition. Prentice Hall
  • ...The revised sample space contains 11 outcomes, but the latter are not equally likely.
  • In general, rules for redefining samples spaces - like going from (x, y's to (x + y)'s are called random variables. As a conceptual framework, random variables are of fundamental importance: they provide a single rubric under which all probability problems may be brought. Even in cases where the original sample space needs no redefinition - that is, where the measurement recorded is the measurement of interests - the concept still applies: we simply take the random variable to be the identify mapping.
  • 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 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 Y 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 S that get mapped into y 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 s and S 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."
  • Associated with each continuous random variable Y is also a probability density function, f_Y(y), but f_Y(y) in this case is not the probability that the random variable Y takes on the value y. Rather, f_Y(y) is a continuous curve having the property that for all a and b,
    • P(a ≤ Y ≤ b) = P({s(∈)S| a ≤ Y(s) ≤ b}) = Integral(a,b). "f_Y(y) dy]

1980

• (Dolciani, Beckenbach, Donnelly, Jurgensen, & Wooton, 1980).
  • "The outcomes form the sample space of the Random Variable"