Random variable
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.[1] The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.[2]
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A random variable's possible values might represent the possible outcomes of a yettobeperformed experiment, or the possible outcomes of a past experiment whose alreadyexisting value is uncertain (for example, because of imprecise measurements or quantum uncertainty).[1] They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a random variable is called a sample space, defined as the set of possible outcomes of a nondeterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
Although the idea was originally introduced by Christiaan Huygens, the first person to think systematically in terms of random variables was Pafnuty Chebyshev.[3][4]
Definition
A random variable is a measurable function from a set of possible outcomes to a measurable space . The technical axiomatic definition requires to be a sample space of a probability triple (see the measuretheoretic definition). A random variable is often denoted by capital roman letters such as , , , .[5][6]
The probability that takes on a value in a measurable set is written as
Standard case
In many cases, is realvalued, i.e. . In some contexts, the term random element (see extensions) is used to denote a random variable not of this form.
When the image (or range) of is countable, the random variable is called a discrete random variable[7]^{:399} and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of . If the image is uncountably infinite (usually an interval) then is called a continuous random variable.[8] In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,[9] a mixture distribution is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
Extensions
The term "random variable" in statistics is traditionally limited to the realvalued case (). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
However, the definition above is valid for any measurable space of values. Thus one can consider random elements of other sets , such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type , or an valued random variable.
This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of nonnumerical data structures. In some cases, it is nonetheless convenient to represent each element of , using one or more real numbers. In this case, a random element may optionally be represented as a vector of realvalued random variables (all defined on the same underlying probability space , which allows the different random variables to covary). For example:
 A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are , , and the position of the 1 indicates the word.
 A random sentence of given length may be represented as a vector of random words.
 A random graph on given vertices may be represented as a matrix of random variables, whose values specify the adjacency matrix of the random graph.
 A random function may be represented as a collection of random variables , giving the function's values at the various points in the function's domain. The are ordinary realvalued random variables provided that the function is realvalued. For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as , and random field is a random function on any set (typically time, space, or a discrete set).
Distribution functions
If a random variable defined on the probability space is given, we can ask questions like "How likely is it that the value of is equal to 2?". This is the same as the probability of the event which is often written as or for short.
Recording all these probabilities of output ranges of a realvalued random variable yields the probability distribution of . The probability distribution "forgets" about the particular probability space used to define and only records the probabilities of various values of . Such a probability distribution can always be captured by its cumulative distribution function
and sometimes also using a probability density function, . In measuretheoretic terms, we use the random variable to "pushforward" the measure on to a measure on . The underlying probability space is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space altogether and just puts a measure on that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on quantile functions for fuller development.
Examples
Discrete random variable
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with nonnegative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum .
In examples such as these, the sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If are countable sets of real numbers, and , then is a discrete distribution function. Here for , for . Taking for instance an enumeration of all rational numbers as , one gets a discrete distribution function that is not a step function or piecewise constant.[7]
Coin toss
The possible outcomes for one coin toss can be described by the sample space . We can introduce a realvalued random variable that models a $1 payoff for a successful bet on heads as follows:
If the coin is a fair coin, Y has a probability mass function given by:
Dice roll
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the twodice case is to take the set of pairs of numbers n_{1} and n_{2} from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable X given by the function that maps the pair to the sum:
and (if the dice are fair) has a probability mass function ƒ_{X} given by:
Continuous random variable
Formally, a continuous random variable is a random variable whose cumulative distribution function is continuous everywhere.[10] There are no "gaps", which would correspond to numbers which have a finite probability of occurring. Instead, continuous random variables almost never take an exact prescribed value c (formally, ) but there is a positive probability that its value will lie in particular intervals which can be arbitrarily small. Continuous random variables usually admit probability density functions (PDF), which characterize their CDF and probability measures; such distributions are also called absolutely continuous; but some continuous distributions are singular, or mixes of an absolutely continuous part and a singular part.
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, X = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any range of values. For example, the probability of choosing a number in [0, 180] is 1⁄2. Instead of speaking of a probability mass function, we say that the probability density of X is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
More formally, given any interval , a random variable is called a "continuous uniform random variable" (CURV) if the probability that it takes a value in a subinterval depends only on the length of the subinterval. This implies that the probability of falling in any subinterval is proportional to the length of the subinterval, that is, if a ≤ c ≤ d ≤ b, one has
where the last equality results from the unitarity axiom of probability. The probability density function of a CURV is given by the indicator function of its interval of support normalized by the interval's length:
Of particular interest is the uniform distribution on the unit interval . Samples of any desired probability distribution can be generated by calculating the quantile function of on a randomlygenerated number distributed uniformly on the unit interval. This exploits properties of cumulative distribution functions, which are a unifying framework for all random variables.
Mixed type
A mixed random variable is a random variable whose cumulative distribution function is neither piecewiseconstant (a discrete random variable) nor everywherecontinuous.[10] It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the CDF will be the weighted average of the CDFs of the component variables.[10]
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of 1⁄2 that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see Lebesgue's decomposition theorem § Refinement. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
Measuretheoretic definition
The most formal, axiomatic definition of a random variable involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigmaalgebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigmaalgebra is used, the Borel σalgebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.[2]
The measuretheoretic definition is as follows.
Let be a probability space and a measurable space. Then an valued random variable is a measurable function , which means that, for every subset , its preimage is measurable; , where .[11] This definition enables us to measure any subset in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of is a possible outcome, a member of is a measurable subset of possible outcomes, the function gives the probability of each such measurable subset, represents the set of values that the random variable can take (such as the set of real numbers), and a member of is a "wellbehaved" (measurable) subset of (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a welldefined probability.
When is a topological space, then the most common choice for the σalgebra is the Borel σalgebra , which is the σalgebra generated by the collection of all open sets in . In such case the valued random variable is called an valued random variable. Moreover, when the space is the real line , then such a realvalued random variable is called simply a random variable.
Realvalued random variables
In this case the observation space is the set of real numbers. Recall, is the probability space. For a real observation space, the function is a realvalued random variable if
This definition is a special case of the above because the set generates the Borel σalgebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that .
Moments
The probability distribution of a 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 , and also called the first moment. In general, is not equal to . Once the "average value" is known, one could then ask how far from this average value the values of typically are, a question that is answered by the variance and standard deviation of a random variable. can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of .
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables , find a collection of functions such that the expectation values fully characterise the distribution of the random variable .
Moments can only be defined for realvalued functions of random variables (or complexvalued, etc.). If the random variable is itself realvalued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function of the random variable. However, even for nonrealvalued random variables, moments can be taken of realvalued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the realvalued function can be constructed; this uses the Iverson bracket, and has the value 1 if has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.
Functions of random variables
A new random variable Y can be defined by applying a real Borel measurable function to the outcomes of a realvalued random variable . That is, . The cumulative distribution function of is then
If function is invertible (i.e., exists, where is 's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain
With the same hypotheses of invertibility of , assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to , in order to obtain[10]
If there is no invertibility of but each admits at most a countable number of roots (i.e., a finite, or countably infinite, number of such that ) then the previous relation between the probability density functions can be generalized with
where , according to the inverse function theorem. The formulas for densities do not demand to be increasing.
In the measuretheoretic, axiomatic approach to probability, if a random variable on and a Borel measurable function , then is also a random variable on , since the composition of measurable functions is also measurable. (However, this is not necessarily true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .
Example 2
Suppose is a random variable with a cumulative distribution
where is a fixed parameter. Consider the random variable Then,
The last expression can be calculated in terms of the cumulative distribution of so
which is the cumulative distribution function (CDF) of an exponential distribution.
Example 3
Suppose is a random variable with a standard normal distribution, whose density is
Consider the random variable We can find the density using the above formula for a change of variables:
In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
The inverse transformation is
and its derivative is
Then,
This is a chisquared distribution with one degree of freedom.
Example 4
Suppose is a random variable with a normal distribution, whose density is
Consider the random variable We can find the density using the above formula for a change of variables:
In this case the change is not monotonic, because every value of has two corresponding values of (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
The inverse transformation is
and its derivative is
Then,
This is a noncentral chisquared distribution with one degree of freedom.
Some properties
 The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
 Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve nonnegativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).
Equivalence of random variables
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
Equality in distribution
If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted ) if they have the same distribution functions:
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.
Almost sure equality
Two random variables X and Y are equal almost surely (denoted ) if, and only if, the probability that they are different is zero:
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
where "ess sup" represents the essential supremum in the sense of measure theory.
Equality
Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space:
This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment is rarely explicitly characterized or even characterizable.
Convergence
A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.
There are various senses in which a sequence of random variables can converge to a random variable . These are explained in the article on convergence of random variables.
See also
 Aleatoricism
 Algebra of random variables
 Event (probability theory)
 Multivariate random variable
 Pairwise independent random variables
 Observable variable
 Random element
 Random function
 Random measure
 Random number generator produces a random value
 Random vector
 Randomness
 Stochastic process
 Relationships among probability distributions
References
Inline citations
 Blitzstein, Joe; Hwang, Jessica (2014). Introduction to Probability. CRC Press. ISBN 9781466575592.
 Steigerwald, Douglas G. "Economics 245A – Introduction to Measure Theory" (PDF). University of California, Santa Barbara. Retrieved April 26, 2013.
 "Christiaan Huygens  Encyclopedia.com". www.encyclopedia.com. Retrieved 20210312.
 George Mackey (July 1980). "HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY  A HISTORICAL SURVEY". Bulletin of the American Mathematical Society. New Series. 3 (1).
 "List of Probability and Statistics Symbols". Math Vault. 20200426. Retrieved 20200821.
 "Random Variables". www.mathsisfun.com. Retrieved 20200821.
 Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN 9780716747734. Archived from the original on 20050209.
 "Random Variables". www.stat.yale.edu. Retrieved 20200821.
 L. Castañeda; V. Arunachalam & S. Dharmaraja (2012). Introduction to Probability and Stochastic Processes with Applications. Wiley. p. 67. ISBN 9781118344941.
 Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
 Fristedt & Gray (1996, page 11)
Literature
 Fristedt, Bert; Gray, Lawrence (1996). A modern approach to probability theory. Boston: Birkhäuser. ISBN 3764338075.
 Kallenberg, Olav (1986). Random Measures (4th ed.). Berlin: Akademie Verlag. ISBN 0123949602. MR 0854102.
 Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Berlin: Springer Verlag. ISBN 0387953132.
 Papoulis, Athanasios (1965). Probability, Random Variables, and Stochastic Processes (9th ed.). Tokyo: McGraw–Hill. ISBN 0071199810.
External links
 "Random variable", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Zukerman, Moshe (2014), Introduction to Queueing Theory and Stochastic Teletraffic Models (PDF), arXiv:1307.2968
 Zukerman, Moshe (2014), Basic Probability Topics (PDF)