Arandom variable (also calledrandom quantity,aleatory variable, orstochastic variable) is a mathematical formalization of a quantity or object which depends onrandom events.[1] The term 'random variable' in its mathematical definition refers to neither randomness nor variability[2] but instead is a mathematicalfunction in which
thedomain is the set of possibleoutcomes in asample space (e.g. the set which are the possible upper sides of a flipped coin heads or tails as the result from tossing a coin); and
therange is ameasurable space (e.g. corresponding to the domain above, the range might be the set if say heads mapped to -1 and mapped to 1). Typically, the range of a random variable is a subset of thereal numbers.
This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions.
Informally, randomness typically represents some fundamental element of chance, such as in the roll of adie; it may also represent uncertainty, such asmeasurement error.[1] However, theinterpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorousaxiomatic setup.
In the formal mathematical language ofmeasure theory, a random variable is defined as ameasurable function from aprobability measure space (called thesample space) to ameasurable space. This allows consideration of thepushforward measure, which is called thedistribution of the random variable; the distribution is thus aprobability measure on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may beindependent.
In many cases, isreal-valued, i.e.. In some contexts, the termrandom element (seeextensions) is used to denote a random variable not of this form.
When theimage (or range) of is finitely or infinitelycountable, the random variable is called adiscrete random variable[5]: 399 and its distribution is adiscrete probability distribution, i.e. can be described by aprobability mass function that assigns a probability to each value in the image of. If the image is uncountably infinite (usually aninterval) then is called acontinuous random variable.[6][7] In the special case that it isabsolutely continuous, its distribution can be described by aprobability 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.[8]
Any random variable can be described by itscumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
The term "random variable" in statistics is traditionally limited to thereal-valued case (). In this case, the structure of the real numbers makes it possible to define quantities such as theexpected value andvariance of a random variable, itscumulative distribution function, and themoments of its distribution.
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.
Arandom graph on given vertices may be represented as a matrix of random variables, whose values specify theadjacency matrix of the random graph.
Arandom 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 real-valued random variables provided that the function is real-valued. For example, astochastic process is a random function of time, arandom vector is a random function of some index set such as, andrandom field is a random function on any set (typically time, space, or a discrete set).
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 outputs of a random variable yields theprobability distribution of. The probability distribution "forgets" about the particular probability space used to define and only records the probabilities of various output values of. Such a probability distribution, if is real-valued, can always be captured by itscumulative distribution function
and sometimes also using aprobability density function,. Inmeasure-theoretic terms, we use the random variable to "push-forward" the measure on to a measure on. The measure is called the "(probability) distribution of" or the "law of".[9] The density, theRadon–Nikodym derivative of with respect to some reference measure on (often, this reference measure is theLebesgue measure in the case of continuous random variables, or thecounting measure in the case of discrete random variables).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 ascorrelation and dependence orindependence based on ajoint 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 onquantile functions for fuller development.
Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their 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.
In examples such as these, thesample 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 function that is not necessarily a step function (piecewise constant).
The possible outcomes for one coin toss can be described by the sample space. We can introduce a real-valued random variable that models a $1 payoff for a successful bet on heads as follows:
If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sumS of the numbers on the two dice, thenS is a discrete random variable whose distribution is described by theprobability mass function plotted as the height of picture columns here.
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 numbersn1 andn2 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 variableX given by the function that maps the pair to the sum:and (if the dice arefair) has a probability mass functionfX given by:
Formally, a continuous random variable is a random variable whosecumulative distribution function iscontinuous everywhere.[10] There are no "gaps", which would correspond to numbers which have a finite probability ofoccurring. Instead, continuous random variablesalmost never take an exact prescribed valuec (formally,) but there is a positive probability that its value will lie in particularintervals which can bearbitrarily small. Continuous random variables usually admitprobability density functions (PDF), which characterize their CDF andprobability measures; such distributions are also calledabsolutely continuous; but some continuous distributions aresingular, 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 anyrange of values. For example, the probability of choosing a number in [0, 180] is1⁄2. Instead of speaking of a probability mass function, we say that the probabilitydensity ofX 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 anyinterval, a random variable is called a "continuous uniform random variable" (CURV) if the probability that it takes a value in asubinterval depends only on the length of the subinterval. This implies that the probability of falling in any subinterval isproportional to thelength of the subinterval, that is, ifa ≤c ≤d ≤b, one has
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; otherwiseX = the value of the spinner as in the preceding example. There is a probability of1⁄2 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; seeLebesgue'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).
The most formal,axiomatic definition of a random variable involvesmeasure theory. Continuous random variables are defined in terms ofsets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. theBanach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed asigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, theBorel 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 orcountably infinite number ofunions and/orintersections of such intervals.[11]
The measure-theoretic definition is as follows.
Let be aprobability space and ameasurable space. Then an-valued random variable is a measurable function, which means that, for every subset, itspreimage is-measurable;, where.[12] 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 "well-behaved" (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 well-defined probability.
When is atopological space, then the most common choice for theσ-algebra is theBorel σ-algebra, which is the σ-algebra 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 real-valued random variable is called simply arandom variable.
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 real-valued random variable if
This definition is a special case of the above because the set 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.
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 ofexpected value of a random variable, denoted, and also called thefirstmoment. 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 thevariance andstandard 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 thedistribution of the random variable.
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 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 acategorical random variableX that can take on thenominal values "red", "blue" or "green", the real-valued function can be constructed; this uses theIverson bracket, and has the value 1 if has the value "green", 0 otherwise. Then, theexpected value and other moments of this function can be determined.
If function is invertible (i.e., exists, where is'sinverse function) and is eitherincreasing or decreasing, then the previous relation can be extended to obtain
With the same hypotheses of invertibility of, assuming alsodifferentiability, the relation between theprobability 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 theprobability density functions can be generalized with
where, according to theinverse function theorem. The formulas for densities do not demand to be increasing.
In the measure-theoretic,axiomatic approach to probability, if a random variable on and aBorel measurable function, then is also a random variable on, since the composition of measurable functionsis also measurable. (However, this is not necessarily true if isLebesgue measurable.[citation needed]) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of.
Consider the random variable We can find the density using the above formula for a change of variables:
In this case the change is notmonotonic, because every value of has two corresponding values of (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
Consider the random variable We can find the density using the above formula for a change of variables:
In this case the change is notmonotonic, 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 probability distribution of the sum of two independent random variables is theconvolution of each of their distributions.
Probability distributions are not avector space—they are not closed underlinear combinations, as these do not preserve non-negativity or total integral 1—but they are closed underconvex combination, thus forming aconvex subset of the space of functions (or measures).
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.
If the sample space is a subset of the real line, random variablesX andY areequal 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 equalmoment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions ofindependent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a definedLaplace transform.
Two random variablesX andY areequalalmost surely (denoted) if, and only if, the probability that they are different iszero:
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
Finally, the two random variablesX andY areequal 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 underlyingmeasure space of theexperiment is rarely explicitly characterized or even characterizable.
Practical difference between notions of equivalence
Since we rarely explicitly construct the probability space underlying a random variable, the difference between these notions of equivalence is somewhat subtle. Essentially, two random variables consideredin isolation are "practically equivalent" if they are equal in distribution -- but once we relate them toother random variables defined on the same probability space, then they only remain "practically equivalent" if they are equal almost surely.
For example, consider the real random variablesA,B,C, andD all defined on the same probability space. Suppose thatA andB are equal almost surely (), butA andC are only equal in distribution (). Then, but in general (not even in distribution). Similarly, we have that the expectation values, but in general. Therefore, two random variables that are equal in distribution (but not equal almost surely) can have differentcovariances with a third random variable.
There are various senses in which a sequence of random variables can converge to a random variable. These are explained in the article onconvergence of random variables.
^George Mackey (July 1980). "Harmonic analysis as the exploitation of symmetry – a historical survey".Bulletin of the American Mathematical Society. New Series.3 (1).
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