Random Variable

Definition: Random Variable

Random Variable

Noun

1. A variable quantity that is random.

Specialty Definitions: Random Variable

Domain	Definitions
Aerospace	A variable characterized by random behavior in assuming its different possible values. Mathematically, it is described by its probability distribution, which specifies the possible values of a random variables together with the probability associated (in an appropriate sense) with each value. A random variable is said to be continuous if its possible values extend over a continuum and discrete if its possible values are separated by finite intervals. Also called variate. See probability theory. (references)
Statistics	In contradistinction to a variable, a variate is a quantity which may take any of the values of a specified set with a specified relative frequency or probability. The variate is therefore often known as a random variable. It is to be regarded as defined, not merely by a set of permissible values like an ordinary mathematical variable, but by an associated frequency(probability)function expressing how often those values appear in the situation under discussion. Source: European Union. (references)
Source: compiled by the editor from various references; see credits.

Top

Specialty Definition: Random variable

(From Wikipedia, the free Encyclopedia)

We can think of a random variable as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, rolling a die and recording the outcome yields a random variable with range {1,2,3,4,5,6}. Picking a random person and measuring their height yields another random variable.

Mathematically, a random variable is defined as a measurable function from a probability space to some measurable space. This measurable space is the space of possible values of the variable, and it is usually taken to be the real numbers with the Borel σ-algebra, and we will always assume this in this encyclopedia, unless otherwise specified.

Distribution functions

If a random variable X:Ω->R defined on the probability space (Ω, P) is given, we can ask questions like "How likely is it that the value of X is bigger than 2?". This is the same as the probability of the event {s in Ω : X(s) > 2} which is often written as P(X > 2) for short.

Recording all these probabilities of output ranges of a real-valued random variable X yields the probability distribution of X. The probability distribution "forgets" about the particular probability space used to define X and only records the probabilities of various values of X. Such a probability distribution can always be captured by its cumulative distribution function

F_X(x) = P(X≤x)

and sometimes also using a probability density function. In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on Ω to a measure dF on R. The underlying probability space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just puts a measure on R that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables.

Functions of random variables

If we have a random variable X on Ω and a measurable function f:R->R, then Y=f(X) will also be a random variable on Ω, since the composition of measurable functions is measurable. The same procedure that allowed one to go from a probability space (Ω,P) to (R,dF_X) can be used to obtain the probability distribution of Y. The cumulative distribution function of Y is

F_Y(y) = Prob(f(X\)≤y).

Example

Let X be a real-valued random variable and let Y = X². Then,

F_Y(y) = Prob(X²≤y).

If y<0, then Prob(X²≤y)=0, so

F_Y(y) = 0 if y<0.

If y≥0, then Prob(X²≤y)=Prob(|X|≤√y)=Prob(-√y≤X≤√y), so

F_Y(y) = F_X(√y)-F_X(-√y) if y≥0.

Moments

The probability distribution of random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[X]. Note that in general, E[f(X)] is not the same as f(E[X]). Once the "average value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable.

Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X, find a collection {f_i} of functions such that the expectation values E[f_i(X)] fully characterize the distribution of the random variable X.

Convergence

Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbers and the central limit theorem.

There are various senses in which a sequence (X_n) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.

Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Random variable."

Top

Synonyms: Random Variable

Synonyms: chance variable (n), stochastic variable (n), variant (n), variate (n). (additional references)

Top

Crosswords: Random Variable

English words defined with "random variable": arithmetic mean ♦ chance variable ♦ expectation, expected value ♦ first moment ♦ modal value, mode ♦ second moment, stationary stochastic process, stochastic, stochastic process, stochastic variable ♦ variance, variant, variate. (references)

Specialty definitions using "random variable": centered random variable, centred random variable, Chi-Square Distribution ♦ extreme value ♦ Gaussian taper ♦ multivariate processes ♦ normal probability density distribution ♦ population mean ♦ simultaneous processes, sonograph ♦ vector processes. (references)

Top

Expressions: Random Variable

Expressions using "random variable": centered random variable ♦ centred random variable. Additional references.
Source: compiled by the editor from various references; see credits.

Top

Frequency of Internet Keywords: Random Variable

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.

Expression	Frequency per Day
random variable	14
discrete random variable	5
function random variable	4
coefficient correlation kendall random variable	3
continuous random variable	2
binomial random variable	2
Source: compiled by the editor from various references; see credits.

Top

Modern Translations: Random Variable

Language		Translations for "random variable"; alternative meanings/domain in parentheses.
Danish		Stokastisk variabel (variate), statistisk variabel (variate). (various references)

Dutch		variabele (variable, variate). (various references)

Finnish		keskitetty satunnaismuuttuja (centered random variable, centred random variable). (various references)

French		variate, variable aléatoire. (various references)

German		Zufallsvariable (aleatory variable, variate), Variate (variate). (various references)

Greek		τυχαία μεταβλητή (variate). (various references)

Italian		variabile aleatoria (chance variable(USA), fortuitous variable, variate), variabile (changeable, changing, different, v., variable, variate, vbl.). (various references)

Japanese Kanji		確率変数 (stochastic variable). (various references)

Japanese Katakana		かくりつへ�"すう (stochastic variable). (various references)

Pig Latin		andomray ariablevay.(various references)

Portuguese		variável aleatória (variate). (various references)

Swedish		centrerad stokastisk variabel (centered random variable, centred random variable). (various references)
Source: compiled by the editor from various translation references.

Top

Anagrams: Random Variable

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-a-a-b-d-e-i-l-m-n-o-r-r-v"
	-4 letters: adenoviral, linerboard.
	-5 letters: abdominal, admirable, adverbial, avoidable, bandolier, bromelain, bromeliad, ealdorman, lamebrain.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.