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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 probability the probability density function of a curv x u a b displaystyle x sim operatorname u a b is given by the indicator function of its interval of support normalized by the interval s length f x x 1 b a a x b 0 otherwise displaystyle f_ x x begin cases displaystyle 1 over b a a leq x leq b 0 text otherwise end cases of particular interest is the uniform distribution on the unit interval 0 1 displaystyle 0 1 samples of any desired probability distribution d displaystyle operatorname d can be generated by calculating the quantile function of d displaystyle operatorname d 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 edit a mixed random variable is a random variable whose cumulative distribution function is neither discrete nor everywhere continuous 10 it can be realized as a mixture 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 is 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 measure theoretic definition edit 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 11 the measure theoretic definition is as follows let ω f p displaystyle omega mathcal f p be a probability space and e e displaystyle e mathcal e a measurable space then an e e displaystyle e mathcal e valued random variable is a measurable function x ω e displaystyle x omega to e which means that for every subset b e displaystyle b in mathcal e its preimage is f displaystyle mathcal f measurable x 1 b f displaystyle x 1 b in mathcal f where x 1 b ω x ω b displaystyle x 1 b omega x omega in b 12 this definition enables us to measure any subset b e displaystyle b in mathcal e in the target space by looking at its preimage which by assumption is measurable in more intuitive terms a member of ω displaystyle omega is a possible outcome a member of f displaystyle mathcal f is a measurable subset of possible outcomes the function p displaystyle p gives the probability of each such measurable subset e displaystyle e represents the set of values that the random variable can take such as the set of real numbers and a member of e displaystyle mathcal e is a well behaved measurable subset of e displaystyle e 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 e displaystyle e is a topological space then the most common choice for the σ algebra e displaystyle mathcal e is the borel σ algebra b e displaystyle mathcal b e which is the σ algebra generated by the collection of all open sets in e displaystyle e in such case the e ...