Random variable

Arandom variable(also calledrandom quantity,aleatory variable,orstochastic variable) is amathematicalformalization of a quantity or object which depends onrandomevents.The term 'random variable' in its mathematical definition refers to neither randomness nor variabilitybut instead is a mathematicalfunctionin which

• thedomainis the set of possibleoutcomesin asample space(e.g. the set$\{H,T\}$which are the possible upper sides of a flipped coin heads$H$or tails$T$as the result from tossing a coin); and
• therangeis ameasurable space(e.g. corresponding to the domain above, the range might be the set$\{-1,1\}$if say heads$H$mapped to -1 and$T$mapped to 1). Typically, the range of a random variable is set ofreal 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 adice;it may also represent uncertainty, such asmeasurement error.However, theinterpretation of probabilityis 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 rigorous axiomatic setup.

In the formal mathematical language ofmeasure theory,a random variable is defined as ameasurable functionfrom aprobability measure space(called thesample space) to ameasurable space.This allows consideration of thepushforward measure,which is called thedistributionof the random variable; the distribution is thus aprobability measureon 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.

It is common to consider the special cases ofdiscrete random variablesandabsolutely continuous random variables,corresponding to whether a random variable is valued in a countable subset or in an interval ofreal numbers.There are other important possibilities, especially in the theory ofstochastic processes,wherein it is natural to considerrandom sequencesorrandom functions.Sometimes arandom variableis taken to be automatically valued in the real numbers, with more general random quantities instead being calledrandom elements.

According toGeorge Mackey,Pafnuty Chebyshevwas the first person "to think systematically in terms of random variables".

Definition

Arandom variable$X$is ameasurable function$X\colon \Omega \to E$from a sample space$\Omega$as a set of possibleoutcomesto ameasurable space$E$.The technical axiomatic definition requires the sample space$\Omega$to be a sample space of aprobability triple$(\Omega,{\mathcal {F}},\operatorname {P} )$(see themeasure-theoretic definition). A random variable is often denoted by capitalRoman letterssuch as$X,Y,Z,T$.

The probability that$X$takes on a value in a measurable set$S\subseteq E$is written as

$\operatorname {P} (X\in S)=\operatorname {P} (\{\ Omega \in \Omega \mid X(\ Omega )\in S\})$.

Standard case

In many cases,$X$isreal-valued,i.e.$E=\mathbb {R}$.In some contexts, the termrandom element(seeextensions) is used to denote a random variable not of this form.

When theimage(or range) of$X$is finitely or infinitelycountable,the random variable is called adiscrete random variableand its distribution is adiscrete probability distribution,i.e. can be described by aprobability mass functionthat assigns a probability to each value in the image of$X$.If the image is uncountably infinite (usually aninterval) then$X$is called acontinuous random variable.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.

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.

Extensions

The term "random variable" in statistics is traditionally limited to thereal-valuedcase ($E=\mathbb {R}$). In this case, the structure of the real numbers makes it possible to define quantities such as theexpected valueandvarianceof a random variable, itscumulative distribution function,and themomentsof its distribution.

However, the definition above is valid for anymeasurable space$E$of values. Thus one can consider random elements of other sets$E$,such as randomBoolean values,categorical values,complex numbers,vectors,matrices,sequences,trees,sets,shapes,manifolds,andfunctions.One may then specifically refer to arandom variable oftype$E$,or an$E$-valued random variable.

This more general concept of arandom elementis particularly useful in disciplines such asgraph theory,machine learning,natural language processing,and other fields indiscrete mathematicsandcomputer science,where one is often interested in modeling the random variation of non-numericaldata structures.In some cases, it is nonetheless convenient to represent each element of$E$,using one or more real numbers. In this case, a random element may optionally be represented as avector of real-valued random variables(all defined on the same underlying probability space$\Omega$,which allows the different random variables tocovary). 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$(1\ 0\ 0\ 0\ \cdots )$,$(0\ 1\ 0\ 0\ \cdots )$,$(0\ 0\ 1\ 0\ \cdots )$and the position of the 1 indicates the word.
• A random sentence of given length$N$may be represented as a vector of$N$random words.
• Arandom graphon$N$given vertices may be represented as a$N\times N$matrix of random variables, whose values specify theadjacency matrixof the random graph.
• Arandom function$F$may be represented as a collection of random variables$F(x)$,giving the function's values at the various points$x$in the function's domain. The$F(x)$are ordinary real-valued random variables provided that the function is real-valued. For example, astochastic processis a random function of time, arandom vectoris a random function of some index set such as$1,2,\ldots,n$,andrandom fieldis a random function on any set (typically time, space, or a discrete set).

Distribution functions

If a random variable$X\colon \Omega \to \mathbb {R}$defined on the probability space$(\Omega,{\mathcal {F}},\operatorname {P} )$is given, we can ask questions like "How likely is it that the value of$X$is equal to 2? ". This is the same as the probability of the event$\{\ Omega:X(\ Omega )=2\}\,\!$which is often written as$P(X=2)\,\!$or$p_{X}(2)$for short.

Recording all these probabilities of outputs of a random variable$X$yields theprobability distributionof$X$.The probability distribution "forgets" about the particular probability space used to define$X$and only records the probabilities of various output values of$X$.Such a probability distribution, if$X$is real-valued, can always be captured by itscumulative distribution function

$F_{X}(x)=\operatorname {P} (X\leq x)$

and sometimes also using aprobability density function,$f_{X}$.Inmeasure-theoreticterms, we use the random variable$X$to "push-forward" the measure$P$on$\Omega$to a measure$p_{X}$on$\mathbb {R}$.The measure$p_{X}$is called the "(probability) distribution of$X$"or the" law of$X$". The density$f_{X}=dp_{X}/d\mu$,theRadon–Nikodym derivativeof$p_{X}$with respect to some reference measure$\mu$on$\mathbb {R}$(often, this reference measure is theLebesgue measurein the case of continuous random variables, or thecounting measurein the case of discrete random variables). The underlying probability space$\Omega$is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such ascorrelation and dependenceorindependencebased on ajoint distributionof two or more random variables on the same probability space. In practice, one often disposes of the space$\Omega$altogether and just puts a measure on$\mathbb {R}$that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article onquantile functionsfor fuller development.

Examples

Discrete random variable

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 his or her 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$\operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots$.

In examples such as these, thesample spaceis 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${\textstyle \{a_{n}\},\{b_{n}\}}$are countable sets of real numbers,${\textstyle b_{n}>0}$and${\textstyle \sum _{n}b_{n}=1}$,then${\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)}$is a discrete distribution function. Here$\delta _{t}(x)=0$for$x<t$,$\delta _{t}(x)=1$for$x\geq t$.Taking for instance an enumeration of all rational numbers as$\{a_{n}\}$,one gets a discrete function that is not necessarily a step function (piecewise constant).

Coin toss

The possible outcomes for one coin toss can be described by the sample space$\Omega =\{{\text{heads}},{\text{tails}}\}$.We can introduce a real-valued random variable$Y$that models a $1 payoff for a successful bet on heads as follows: $$Y(\ Omega )={\begin{cases}1,&{\text{if }}\ Omega ={\text{heads}},\\[6pt]0,&{\text{if }}\ Omega ={\text{tails}}.\end{cases}}$$

If the coin is afair coin,Yhas aprobability mass function$f_{Y}$given by: $$f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}$$

Dice roll

If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sumSof the numbers on the two dice, thenSis a discrete random variable whose distribution is described by theprobability mass functionplotted 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 numbersn₁andn₂from {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 variableXgiven by the function that maps the pair to the sum: $$X((n_{1},n_{2}))=n_{1}+n_{2}$$ and (if the dice arefair) has a probability mass functionf_Xgiven by: $$f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}$$

Continuous random variable

Formally, a continuous random variable is a random variable whosecumulative distribution functioniscontinuouseverywhere.There are no "gaps",which would correspond to numbers which have a finite probability ofoccurring.Instead, continuous random variablesalmost nevertake an exact prescribed valuec(formally,${\textstyle \forall c\in \mathbb {R}:\;\Pr(X=c)=0}$) but there is a positive probability that its value will lie in particularintervalswhich 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 anyrangeof 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 probabilitydensityofXis 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${\textstyle I=[a,b]=\{x\in \mathbb {R}:a\leq x\leq b\}}$,a random variable$X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]$is called a "continuous uniformrandom variable "(CURV) if the probability that it takes a value in asubintervaldepends only on the length of the subinterval. This implies that the probability of$X_{I}$falling in any subinterval$[c,d]\subseteq [a,b]$isproportionalto thelengthof the subinterval, that is, ifa≤c≤d≤b,one has

$$\Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}$$

where the last equality results from theunitarity axiomof probability. Theprobability density functionof a CURV$X\sim \operatorname {U} [a,b]$is given by theindicator functionof its interval ofsupportnormalized by the interval's length:$$f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}$$Of particular interest is the uniform distribution on theunit interval$[0,1]$.Samples of any desiredprobability distribution$\operatorname {D}$can be generated by calculating thequantile functionof$\operatorname {D}$on arandomly-generated numberdistributed uniformly on the unit interval. This exploitsproperties of cumulative distribution functions,which are a unifying framework for all random variables.

Mixed type

Amixed random variableis a random variable whosecumulative distribution functionis neitherdiscretenoreverywhere-continuous.It can be realized as a mixture of a discrete random variable and a continuous random variable; in which case theCDFwill 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; otherwiseX= the value of the spinner as in the preceding example. There is a probability of1⁄2that 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).

Measure-theoretic definition

The most formal,axiomaticdefinition of a random variable involvesmeasure theory.Continuous random variables are defined in terms ofsetsof 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-algebrato 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 infinitenumber ofunionsand/orintersectionsof such intervals.

The measure-theoretic definition is as follows.

Let$(\Omega,{\mathcal {F}},P)$be aprobability spaceand$(E,{\mathcal {E}})$ameasurable space.Then an$(E,{\mathcal {E}})$-valued random variableis a measurable function$X\colon \Omega \to E$,which means that, for every subset$B\in {\mathcal {E}}$,itspreimageis${\mathcal {F}}$-measurable;$X^{-1}(B)\in {\mathcal {F}}$,where$X^{-1}(B)=\{\ Omega:X(\ Omega )\in B\}$.This definition enables us to measure any subset$B\in {\mathcal {E}}$in the target space by looking at its preimage, which by assumption is measurable.

In more intuitive terms, a member of$\Omega$is a possible outcome, a member of${\mathcal {F}}$is a measurable subset of possible outcomes, the function$P$gives the probability of each such measurable subset,$E$represents the set of values that the random variable can take (such as the set of real numbers), and a member of${\mathcal {E}}$is a "well-behaved" (measurable) subset of$E$(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$E$is atopological space,then the most common choice for theσ-algebra${\mathcal {E}}$is theBorel σ-algebra${\mathcal {B}}(E)$,which is the σ-algebra generated by the collection of all open sets in$E$.In such case the$(E,{\mathcal {E}})$-valued random variable is called an$E$-valued random variable.Moreover, when the space$E$is the real line$\mathbb {R}$,then such a real-valued random variable is called simply arandom variable.

Real-valued random variables

In this case the observation space is the set of real numbers. Recall,$(\Omega,{\mathcal {F}},P)$is the probability space. For a real observation space, the function$X\colon \Omega \rightarrow \mathbb {R}$is a real-valued random variable if

$\{\ Omega:X(\ Omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R}.$

This definition is a special case of the above because the set$\{(-\infty,r]:r\in \mathbb {R} \}$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$\{\ Omega:X(\ Omega )\leq r\}=X^{-1}((-\infty,r])$.

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 ofexpected valueof a random variable, denoted$\operatorname {E} [X]$,and also called thefirstmoment.In general,$\operatorname {E} [f(X)]$is not equal to$f(\operatorname {E} [X])$.Once the "average value" is known, one could then ask how far from this average value the values of$X$typically are, a question that is answered by thevarianceandstandard deviationof a random variable.$\operatorname {E} [X]$can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of$X$.

Mathematically, this is known as the (generalised)problem of moments:for a given class of random variables$X$,find a collection$\{f_{i}\}$of functions such that the expectation values$\operatorname {E} [f_{i}(X)]$fully characterise thedistributionof the random variable$X$.

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$f(X)=X$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 acategoricalrandom variableXthat can take on thenominalvalues "red", "blue" or "green", the real-valued function$[X={\text{green}}]$can be constructed; this uses theIverson bracket,and has the value 1 if$X$has the value "green", 0 otherwise. Then, theexpected valueand other moments of this function can be determined.

Functions of random variables

A new random variableYcan be defined byapplyinga realBorel measurable function$g\colon \mathbb {R} \rightarrow \mathbb {R}$to the outcomes of areal-valuedrandom variable$X$.That is,$Y=g(X)$.Thecumulative distribution functionof$Y$is then

$F_{Y}(y)=\operatorname {P} (g(X)\leq y).$

If function$g$is invertible (i.e.,$h=g^{-1}$exists, where$h$is$g$'sinverse function) and is eitherincreasing or decreasing,then the previous relation can be extended to obtain

$F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}$

With the same hypotheses of invertibility of$g$,assuming alsodifferentiability,the relation between theprobability density functionscan be found by differentiating both sides of the above expression with respect to$y$,in order to obtain

$f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.$

If there is no invertibility of$g$but each$y$admits at most a countable number of roots (i.e., a finite, or countably infinite, number of$x_{i}$such that$y=g(x_{i})$) then the previous relation between theprobability density functionscan be generalized with

$f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|$

where$x_{i}=g_{i}^{-1}(y)$,according to theinverse function theorem.The formulas for densities do not demand$g$to be increasing.

In the measure-theoretic,axiomatic approachto probability, if a random variable$X$on$\Omega$and aBorel measurable function$g\colon \mathbb {R} \rightarrow \mathbb {R}$,then$Y=g(X)$is also a random variable on$\Omega$,since the composition of measurable functionsis also measurable.(However, this is not necessarily true if$g$isLebesgue measurable.) The same procedure that allowed one to go from a probability space$(\Omega,P)$to$(\mathbb {R},dF_{X})$can be used to obtain the distribution of$Y$.

Example 1

Let$X$be a real-valued,continuous random variableand let$Y=X^{2}$.

$F_{Y}(y)=\operatorname {P} (X^{2}\leq y).$

If$y<0$,then$P(X^{2}\leq y)=0$,so

$F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.$

If$y\geq 0$,then

$\operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),$

so

$F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.$

Example 2

Suppose$X$is a random variable with a cumulative distribution

$F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}$

where$\theta >0$is a fixed parameter. Consider the random variable$Y=\mathrm {log} (1+e^{-X}).$Then,

$F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,$

The last expression can be calculated in terms of the cumulative distribution of$X,$so

${\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\[5pt]&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\[5pt]&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\[5pt]&=1-e^{-y\theta }.\end{aligned}}$

which is thecumulative distribution function(CDF) of anexponential distribution.

Example 3

Suppose$X$is a random variable with astandard normal distribution,whose density is

$f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.$

Consider the random variable$Y=X^{2}.$We can find the density using the above formula for a change of variables:

$f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.$

In this case the change is notmonotonic,because every value of$Y$has two corresponding values of$X$(one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,

$f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.$

The inverse transformation is

$x=g^{-1}(y)={\sqrt {y}}$

and its derivative is

${\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.$

Then,

$f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.$

This is achi-squared distributionwith onedegree of freedom.

Example 4

Suppose$X$is a random variable with anormal distribution,whose density is

$f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.$

Consider the random variable$Y=X^{2}.$We can find the density using the above formula for a change of variables:

$f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.$

In this case the change is notmonotonic,because every value of$Y$has two corresponding values of$X$(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:

$f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.$

The inverse transformation is

$x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}$

and its derivative is

${\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.$

Then,

$f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).$

This is anoncentral chi-squared distributionwith onedegree of freedom.

Some properties

• The probability distribution of the sum of two independent random variables is theconvolutionof 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 subsetof the space of functions (or measures).

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 variablesXandYareequal in distribution(denoted$X{\stackrel {d}{=}}Y$) if they have the same distribution functions:

$\operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.$

To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equalmoment generating functionshave 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.

Almost sure equality

Two random variablesXandYareequalalmost surely(denoted$X\;{\stackrel {\text{a.s.}}{=}}\;Y$) if, and only if, the probability that they are different iszero:

$\operatorname {P} (X\neq Y)=0.$

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

$d_{\infty }(X,Y)=\operatorname {ess} \sup _{\ Omega }|X(\ Omega )-Y(\ Omega )|,$

where "ess sup" represents theessential supremumin the sense ofmeasure theory.

Equality

Finally, the two random variablesXandYareequalif they are equal as functions on their measurable space:

$X(\ Omega )=Y(\ Omega )\qquad {\hbox{for all }}\ Omega.$

This notion is typically the least useful in probability theory because in practice and in theory, the underlyingmeasure spaceof theexperimentis rarely explicitly characterized or even characterizable.

Convergence

Main article:Convergence of random variables

A significant theme in mathematical statistics consists of obtaining convergence results for certainsequencesof random variables; for instance thelaw of large numbersand thecentral limit theorem.

There are various senses in which a sequence$X_{n}$of random variables can converge to a random variable$X$.These are explained in the article onconvergence of random variables.

See also

• Aleatoricism
• Algebra of random variables
• Event (probability theory)
• Multivariate random variable
• Pairwise independent random variables
• Observable variable
• Random compact set
• Random element
• Random function
• Random measure
• Random number generator
• Random variate
• Random vector
• Randomness
• Stochastic process
• Relationships among probability distributions

References

Literature

• Fristedt, Bert; Gray, Lawrence (1996).A modern approach to probability theory.Boston: Birkhäuser.ISBN3-7643-3807-5.
• Billingsley, Patrick (1995).Probability and Measure.New York: Wiley.ISBN8126517719.
• Kallenberg, Olav(1986).Random Measures(4th ed.). Berlin:Akademie Verlag.ISBN0-12-394960-2.MR0854102.
• Kallenberg, Olav (2001).Foundations of Modern Probability(2nd ed.). Berlin:Springer Verlag.ISBN0-387-95313-2.
• Papoulis, Athanasios(1965).Probability, Random Variables, and Stochastic Processes(9th ed.). Tokyo:McGraw–Hill.ISBN0-07-119981-0.

External links

• "Random variable",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Zukerman, Moshe (2014),Introduction to Queueing Theory and Stochastic Teletraffic Models(PDF),arXiv:1307.2968
• Zukerman, Moshe (2014),Basic Probability Topics(PDF)