Random variable
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In probability and statistics, a random variable, random quantity, aleatory variable or stochastic variable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).^{[1]}^{:391} A random variable can take on a set of possible different values (similarly to other mathematical variables), each with an associated probability, in contrast to other mathematical variables.
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, due to imprecise measurements or quantum uncertainty). 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.
The mathematical function describing the possible values of a random variable and their associated probabilities is known as a probability distribution. Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function, characteristic of a probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of a probability distribution; or a mixture of both types. 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.
The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a function defined on a sample space whose outputs are numerical values.^{[2]}
Contents
Definition
A random variable is a measurable function from the set of possible outcomes to some set . The technical axiomatic definition requires to be a probability space and to be a measurable space (see Measuretheoretic definition).
Note that although is usually a realvalued function (), it does not return a probability. The probabilities of different outcomes or sets of outcomes (events) are already given by the probability measure with which is equipped. Rather, describes some numerical property that outcomes in may have. E.g. the number of heads in a random collection of coin flips; the height of a random person. The probability that takes value is the measure of the set of outcomes , denoted .
Standard case
Usually . Otherwise the term random element is used (see Extensions).
When the image (or range) of is finite or countably infinite, the random variable is called a discrete random variable^{[1]}^{:399} and its distribution can be described by a probability mass function which assigns a probability to each value in the image of . If the image is uncountably infinite then is called a continuous random variable. 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,^{[3]} for example a mixture distribution. 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 (). This ensures that it is 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 (1 0 0 0 ...), (0 1 0 0 ...), (0 0 1 0 ...), 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).
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 PMF(0) + PMF(2) + PMF(4) + ...
In examples such as these, the sample space (the set of all possible persons) 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.
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:
Die 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
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.
Mixed type
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 probability of the last example.
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 function which is measurable. The latter means that, for every subset , its preimage where .^{[4]} This definition enables us to measure any subset in the target space by looking at its preimage, which by assumption is measurable.
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 the valued random variable. Moreover, when space is the real line , then such a realvalued random variable is called simply the random variable.
Realvalued random variables
In this case the observation space is the set of real numbers. Recall, is the probability space. For 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 .
Distribution functions of random variables
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.
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 E[X], and also called the first moment. In general, E[f(X)] is not equal to 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. E[X] can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of X.
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 characterise the distribution of the random variable X.
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 X 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 X. The cumulative distribution function of is
If function g is invertible, i.e. g^{−1} exists, and is either increasing or decreasing, then the previous relation can be extended to obtain
and, again with the same hypotheses of invertibility of g, assuming also differentiability, we can find the relation between the probability density functions by differentiating both sides with respect to y, in order to obtain
If there is no invertibility of g but each y admits at most a countable number of roots (i.e. a finite, or countably infinite, number of x_{i} such that y = g(x_{i})) then the previous relation between the probability density functions can be generalized with
where x_{i} = g_{i}^{−1}(y). The formulas for densities do not demand g to be increasing.
In the measuretheoretic, axiomatic approach to probability, if we have a random variable on and a Borel measurable function , then will also be a random variable on , since the composition of measurable functions is also measurable. (However, this is not 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 1
Let X be a realvalued, continuous random variable and let Y = X^{2}.
If y < 0, then P(X^{2} ≤ y) = 0, so
If y ≥ 0, then
so
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 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.
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:
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 i.i.d. 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 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:
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 (X_{n}) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.
See also
References
 ↑ ^{1.0} ^{1.1} Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN 9780716747734.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ ^{2.0} ^{2.1} Steigerwald, Douglas G. "Economics 245A – Introduction to Measure Theory" (PDF). University of California, Santa Barbara. Retrieved April 26, 2013.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ L. Castañeda, V. Arunachalam, and S. Dharmaraja (2012). Introduction to Probability and Stochastic Processes with Applications. Wiley. p. 67.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Fristedt & Gray (1996, page 11)
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (February 2012)

Literature
 Fristedt, Bert; Gray, Lawrence (1996). A modern approach to probability theory. Boston: Birkhäuser. ISBN 3764338075.CS1 maint: ref=harv (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Kallenberg, Olav (1986). Random Measures (4th ed.). Berlin: Akademie Verlag. ISBN 0123949602. MR 0854102.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Berlin: Springer Verlag. ISBN 0387953132.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Papoulis, Athanasios (1965). Probability, Random Variables, and Stochastic Processes (9th ed.). Tokyo: McGraw–Hill. ISBN 0071199810.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
External links
 Hazewinkel, Michiel, ed. (2001), "Random variable", Encyclopedia of Mathematics, Springer, ISBN 9781556080104<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Zukerman, Moshe (2014), Introduction to Queueing Theory and Stochastic Teletraffic Models (PDF)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 Zukerman, Moshe (2014), Basic Probability Topics (PDF)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>