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. 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.
A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of 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.
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 non-deterministic 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.
Definition.
A random variable formula_1 is a measurable function formula_2 from a set of possible outcomes formula_3 to a measurable space formula_4. The technical axiomatic definition requires formula_5 to be a sample space of a probability triple formula_6 (see the measure-theoretic definition). A random variable is often denoted by capital roman letters such as formula_1, formula_8, formula_9, formula_10.
The probability that formula_1 takes on a value in a measurable set formula_12 is written as
Standard case.
In many cases, formula_1 is real-valued, i.e. formula_15. 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 formula_1 is countable, the random variable is called a discrete random variable 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 formula_1. If the image is uncountably infinite (usually an interval) then formula_1 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, 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 real-valued case (formula_19). 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 formula_20 of values. Thus one can consider random elements of other sets formula_20, 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 formula_20", or an "formula_20-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 non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of formula_20, using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space formula_5, which allows the different random variables to covary). For example:
Distribution functions.
If a random variable formula_38 defined on the probability space formula_6 is given, we can ask questions like "How likely is it that the value of formula_1 is equal to 2?". This is the same as the probability of the event formula_41 which is often written as formula_42 or formula_43 for short.
Recording all these probabilities of output ranges of a real-valued random variable formula_1 yields the probability distribution of formula_1. The probability distribution "forgets" about the particular probability space used to define formula_1 and only records the probabilities of various values of formula_1. Such a probability distribution can always be captured by its cumulative distribution function
and sometimes also using a probability density function, formula_49. In measure-theoretic terms, we use the random variable formula_1 to "push-forward" the measure formula_51 on formula_5 to a measure formula_49 on formula_54.
The underlying probability space formula_5 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 formula_5 altogether and just puts a measure on formula_54 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 non-negative 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 formula_58.
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 formula_59 are countable sets of real numbers, formula_60 and formula_61, then formula_62 is a discrete distribution function. Here formula_63 for formula_64, formula_65 for formula_66. Taking for instance an enumeration of all rational numbers as formula_67, one gets a discrete distribution function that is not a step function or piecewise constant.
Coin toss.
The possible outcomes for one coin toss can be described by the sample space formula_68. We can introduce a real-valued random variable formula_8 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 formula_71 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 two-dice 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. 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, formula_75) 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 . 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 formula_76, a random variable formula_77 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 formula_78 falling in any subinterval formula_79 is proportional to the length of the subinterval, that is, if , one has
formula_80
where the last equality results from the unitarity axiom of probability. The probability density function of a CURV formula_81 is given by the indicator function of its interval of support normalized by the interval's length: formula_82Of particular interest is the uniform distribution on the unit interval formula_83. Samples of any desired probability distribution formula_84 can be generated by calculating the quantile function of formula_84 on a randomly-generated 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 piecewise-constant (a discrete random variable) nor everywhere-continuous. It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the will be the weighted average of the CDFs of the component variables.
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 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 . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
Measure-theoretic 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 sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, 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.
The measure-theoretic definition is as follows.
Let formula_86 be a probability space and formula_87 a measurable space. Then an formula_87-valued random variable is a measurable function formula_89, which means that, for every subset formula_90, its preimage formula_91 where formula_92. This definition enables us to measure any subset formula_93 in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of formula_5 is a possible outcome, a member of formula_95 is a measurable subset of possible outcomes, the function formula_51 gives the probability of each such measurable subset, formula_20 represents the set of values that the random variable can take (such as the set of real numbers), and a member of formula_98 is a "well-behaved" (measurable) subset of formula_20 (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 well-defined probability.
When formula_20 is a topological space, then the most common choice for the σ-algebra formula_98 is the Borel σ-algebra formula_102, which is the σ-algebra generated by the collection of all open sets in formula_20. In such case the formula_87-valued random variable is called an formula_20-valued random variable. Moreover, when the space formula_20 is the real line formula_54, then such a real-valued random variable is called simply a random variable.
Real-valued random variables.
In this case the observation space is the set of real numbers. Recall, formula_86 is the probability space. For a real observation space, the function formula_109 is a real-valued random variable if
This definition is a special case of the above because the set formula_111 generates the Borel σ-algebra 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 formula_112.
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 formula_113, and also called the first moment. In general, formula_114 is not equal to formula_115. Once the "average value" is known, one could then ask how far from this average value the values of formula_1 typically are, a question that is answered by the variance and standard deviation of a random variable. formula_113 can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of formula_1.
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables formula_1, find a collection formula_120 of functions such that the expectation values formula_121 fully characterise the distribution of the random variable formula_1.
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function formula_123 of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable "X" that can take on the nominal values "red", "blue" or "green", the real-valued function formula_124 can be constructed; this uses the Iverson bracket, and has the value 1 if formula_1 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 formula_126 to the outcomes of a real-valued random variable formula_1. That is, formula_128. The cumulative distribution function of formula_8 is then
If function formula_131 is invertible (i.e., formula_132 exists, where formula_133 is formula_131'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 formula_131, assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to formula_137, in order to obtain
If there is no invertibility of formula_131 but each formula_137 admits at most a countable number of roots (i.e., a finite, or countably infinite, number of formula_141 such that formula_142) then the previous relation between the probability density functions can be generalized with
where formula_144, according to the inverse function theorem. The formulas for densities do not demand formula_131 to be increasing.
In the measure-theoretic, axiomatic approach to probability, if a random variable formula_1 on formula_5 and a Borel measurable function formula_126, then formula_149 is also a random variable on formula_5, since the composition of measurable functions is also measurable. (However, this is not necessarily true if formula_131 is Lebesgue measurable.) The same procedure that allowed one to go from a probability space formula_152 to formula_153 can be used to obtain the distribution of formula_8.
Example 1.
Let formula_1 be a real-valued, continuous random variable and let formula_156.
If formula_158, then formula_159, so
If formula_161, then
so
Example 2.
Suppose formula_1 is a random variable with a cumulative distribution
where formula_166 is a fixed parameter. Consider the random variable formula_167 Then,
The last expression can be calculated in terms of the cumulative distribution of formula_169 so
which is the cumulative distribution function (CDF) of an exponential distribution.
Example 3.
Suppose formula_1 is a random variable with a standard normal distribution, whose density is
Consider the random variable formula_173 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 formula_8 has two corresponding values of formula_1 (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 chi-squared distribution with one degree of freedom.
Example 4.
Suppose formula_1 is a random variable with a normal distribution, whose density is
Consider the random variable formula_173 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 formula_8 has two corresponding values of formula_1 (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 chi-squared 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 formula_191) 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 formula_193) 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 formula_197 of random variables can converge to a random variable formula_1. These are explained in the article on convergence of random variables.