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of such interpretational difficulties and can be based upon a rigorous axiomatic setup in the formal mathematical language of measure theory a random variable is defined as a measurable function from a probability measure space called the sample space to a measurable space this allows consideration of the pushforward measure which is called the distribution of the random variable the distribution is thus a probability 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 be independent it is common to consider the special cases of discrete random variables and absolutely continuous random variables corresponding to whether a random variable is valued in a countable subset or in an interval of real numbers there are other important possibilities especially in the theory of stochastic processes wherein it is natural to consider random sequences or random functions sometimes a random variable is taken to be automatically valued in the real numbers with more general random quantities instead being called random elements a random variate is a particular outcome or realization of a random variable according to george mackey pafnuty chebyshev was the first person to think systematically in terms of random variables 3 definition edit a random variable x displaystyle x is a measurable function x ω e displaystyle x colon omega to e from a sample space ω displaystyle omega as a set of possible outcomes to a measurable space e displaystyle e for the measurability of x displaystyle x to be meaningful the sample space ω displaystyle omega needs to belong to a probability triple ω f p displaystyle omega mathcal f operatorname p see the measure theoretic definition a random variable is often denoted by capital roman letters such as x y z t displaystyle x y z t 4 the probability that x displaystyle x takes on a value in a measurable set s e displaystyle s subseteq e is written as p x s p ω ω x ω s displaystyle operatorname p x in s operatorname p omega in omega mid x omega in s standard case edit in many cases x displaystyle x is real valued i e e r displaystyle e mathbb r 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 x displaystyle x is finite or countably infinite the random variable is called a discrete random variable 5 399 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 x displaystyle x if the image is uncountably infinite usually an interval then x displaystyle x is called a continuous random variable 6 7 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 8 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 edit the term random variable in statistics is traditionally limited to the real valued case e r displaystyle e mathbb r 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 e displaystyle e of values thus one can consider random elements of other sets e displaystyle e 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 e displaystyle e or an e displaystyle e 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 e displaystyle e 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 ω displaystyle omega 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 displaystyle 1 0 0 0 cdots 0 1 0 0 displaystyle 0 1 0 0 cdots 0 0 1 0 displaystyle 0 0 1 0 cdots and the position of the 1 indicates the word a random sentence of given length n displaystyle n may be represented as a vector of n displaystyle n random words a random graph on n displaystyle n given vertices may be represented as a n n displaystyle n times n matrix of random variables whose values specify the adjacency matrix of the random graph a random function f displaystyle f may be represented as a collection of random variables f x displaystyle f x giving the function s values at the various points x displaystyle x in the function s domain the f x displaystyle f x are ordinary real valued random variables provided that the function is real valued for example a stochastic process is a random function of time a random vector is a random function of some index set such as 1 2 n displaystyle 1 2 ldots n and a random field is a random function on any set typically time space or a discrete set distribution functions edit if a random variable x ω r displaystyle x colon omega to mathbb r defined on the probability space ω f p displaystyle omega mathcal f operatorname p is given we can ask questions like how likely is it that the value of x displaystyle x is equal to 2 this is the same as the probability of the event ω x ω 2 displaystyle omega x omega 2 which is often written as p x 2 displaystyle p x 2 or p x 2 displaystyle p_ x 2 for short recording all these probabilities of outputs of a random variable x displaystyle x yields the probability distribution of x displaystyle x the probability distribution forgets about the particular probability space used to define x displaystyle x and only records the probabilities of various output values of x displaystyle x such a probability distribution if x displaystyle x is real valued can always be captured by its cumulative distribution function f x x p x x displaystyle f_ x x operatorname p x leq x and sometimes also using a probability density function f x displaystyle f_ x in measure theoretic terms we use the random variable x displaystyle x to push forward the measure p displaystyle p on ω displaystyle omega to a measure p x displaystyle p_ x on r displaystyle mathbb r the measure p x displaystyle p_ x is called the probability distribution of x displaystyle x or the law of x displaystyle x 9 the density f x d p x d μ displaystyle f_ x dp_ x d mu the radon nikodym derivative of p x displaystyle p_ x with respect to some reference measure μ displaystyle mu on r displaystyle mathbb r often this reference measure is the lebesgue measure in the case of continuous random variables or the counting measure in the case of discrete random variables the underlying probability space ω displaystyle omega 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 ω displaystyle omega altogether and just puts a measure on r displaystyle mathbb r 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 edit discrete random variable edit 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 pmf 0 pmf 2 pmf 4 displaystyle operatorname pmf 0 operatorname pmf 2 operatorname pmf 4 cdots 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 a n b n textstyle a_ n b_ n are countable sets of real numbers b n 0 textstyle b_ n 0 and n b n 1 displaystyle textstyle sum _ n b_ n 1 then f n b n δ a n x textstyle f sum _ n b_ n delta _ a_ n x is a discrete distribution function here δ t x 0 displaystyle delta _ t x 0 for x t displaystyle x t δ t x 1 displaystyle delta _ t x 1 for x t displaystyle x geq t taking for instance an enumeration of all rational numbers as a n displaystyle a_ n one gets a discrete function that is not necessarily a step function piecewise constant coin toss edit the possible outcomes for one coin toss can be described by the sample space ω heads tails displaystyle omega text heads text tails we can introduce a real valued random variable y displaystyle y that models a 1 payoff for a successful bet on heads as follows y ω 1 if ω heads 0 if ω tails displaystyle y omega begin cases 1 text if omega text heads 6pt 0 text if omega text tails end cases if the coin is a fair coin y displaystyle y has a probability mass function f y displaystyle f_ y given by f y y 1 2 if y 1 1 2 if y 0 displaystyle f_ y y begin cases tfrac 1 2 text if y 1 6pt tfrac 1 2 text if y 0 end cases dice roll edit if the sample space is the set of possible numbers rolled on two dice and the random variable of interest is the sum s of the numbers on the two dice then s is a discrete random variable whose distribution is described by the probability 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 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 x n 1 n 2 n 1 n 2 displaystyle x n_ 1 n_ 2 n_ 1 n_ 2 and if the dice are fair has a probability mass function f x given by f x s min s 1 13 s 36 for s 2 3 4 5 6 7 8 9 10 11 12 displaystyle f_ x s frac min s 1 13 s 36 text for s in 2 3 4 5 6 7 8 9 10 11 12 continuous random variable edit 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 c r pr x c 0 textstyle forall c in mathbb r pr x c 0 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 i a b x r a x b textstyle i a b x in mathbb r a leq x leq b a random variable x i u i u a b displaystyle x_ i sim operatorname u i operatorname u a b 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 x i displaystyle x_ i falling in any subinterval c d a b displaystyle c d subseteq a b is proportional to the length of the subinterval that is if a c d b one has pr x i c d d c b a displaystyle pr left x_ i in c d right frac d c b a where the last equality results from the unitarity axiom of pro...
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