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n 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 e displaystyle e mathcal e valued random variable is called an e displaystyle e valued random variable moreover when the space e displaystyle e is the real line r displaystyle mathbb r then such a real valued random variable is called simply a random variable note that we are not giving r displaystyle mathbb r the usual lebesgue σ displaystyle sigma algebra which is the completion of the borel σ displaystyle sigma algebra this choice allows for more measurable functions f ω r displaystyle f omega to mathbb r and makes it easier to check that a function f ω r displaystyle f omega to mathbb r is measurable as we only need to check that preimages of open sets are measurable real valued random variables edit in this case the observation space is the set of real numbers recall ω f p displaystyle omega mathcal f p is the probability space for a real observation space the function x ω r displaystyle x colon omega rightarrow mathbb r is a real valued random variable if ω x ω r f r r displaystyle omega x omega leq r in mathcal f qquad forall r in mathbb r this definition is a special case of the above because the set r r r displaystyle infty r r in mathbb r 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 ω x ω r x 1 r displaystyle omega x omega leq r x 1 infty r moments edit 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 displaystyle operatorname e x and also called the first moment in general e f x displaystyle operatorname e f x is not equal to f e x displaystyle f operatorname e x once the average value is known one could then ask how far from this average value the values of x displaystyle x typically are a question that is answered by the variance and standard deviation of a random variable e x displaystyle operatorname e x can be viewed intuitively as an average obtained from an infinite population the members of which are particular evaluations of x displaystyle x mathematically this is known as the generalised problem of moments for a given class of random variables x displaystyle x find a collection f i displaystyle f_ i of functions such that the expectation values e f i x displaystyle operatorname e f_ i x fully characterise the distribution of the random variable x displaystyle x 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 f x x displaystyle f x x 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 x green displaystyle x text green can be constructed this uses the iverson bracket and has the value 1 if x displaystyle x has the value green 0 otherwise then the expected value and other moments of this function can be determined functions of random variables edit a new random variable y displaystyle y can be defined by applying a real borel measurable function g r r displaystyle g colon mathbb r rightarrow mathbb r to the outcomes of a real valued random variable x displaystyle x that is y g x displaystyle y g x the cumulative distribution function of y displaystyle y is then f y y p g x y displaystyle f_ y y operatorname p g x leq y if function g displaystyle g is invertible i e h g 1 displaystyle h g 1 exists where h displaystyle h is g displaystyle g s inverse function and is either increasing or decreasing then the previous relation can be extended to obtain f y y p g x y p x h y f x h y if h g 1 increasing p x h y 1 f x h y if h g 1 decreasing displaystyle f_ y y operatorname p g x leq y begin cases operatorna...
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