Random variable

Standard case

In many cases, ${\displaystyle X}$ is real-valued, i.e. ${\displaystyle E=\mathbb {R} }$. 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 ${\displaystyle X}$ 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 ${\displaystyle X}$. If the image is uncountably infinite (usually an interval) then ${\displaystyle X}$ 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

The term "random variable" in statistics is traditionally limited to the real-valued case (⁠${\displaystyle E=\mathbb {R} }$⁠). 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 ${\displaystyle E}$ of values. Thus one can consider random elements of other sets ${\displaystyle E}$, 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 ${\displaystyle E}$, or an ${\displaystyle E}$-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 ${\displaystyle E}$, 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 ${\displaystyle \Omega }$, 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 ${\displaystyle (1\ 0\ 0\ 0\ \cdots )}$, ${\displaystyle (0\ 1\ 0\ 0\ \cdots )}$, ${\displaystyle (0\ 0\ 1\ 0\ \cdots )}$ and the position of the 1 indicates the word.
• A random sentence of given length ${\displaystyle N}$ may be represented as a vector of ${\displaystyle N}$ random words.
• A random graph on ${\displaystyle N}$ given vertices may be represented as a ${\displaystyle N\times N}$ matrix of random variables, whose values specify the adjacency matrix of the random graph.
• A random function ${\displaystyle F}$ may be represented as a collection of random variables ${\displaystyle F(x)}$, giving the function's values at the various points ${\displaystyle x}$ in the function's domain. The ${\displaystyle F(x)}$ 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 ${\displaystyle 1,2,\ldots ,n}$, and a random field is a random function on any set (typically time, space, or a discrete set).

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 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 ⁠${\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }$⁠.

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

Dice roll

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 n₁ and n₂ 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: $${\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}}$$ and (if the dice are fair) has a probability mass function f_X given by: $${\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}$$

Continuous random variable

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, ${\textstyle \forall c\in \mathbb {R}$ :\;\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 ${\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}}$, a random variable ${\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}$ 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 ${\displaystyle X_{I}}$ falling in any subinterval ${\displaystyle [c,d]\subseteq [a,b]}$ is proportional to the length of the subinterval, that is, if a ≤ c ≤ d ≤ b, one has $${\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}}$$ where the last equality results from the unitarity axiom of probability. The probability density function of a CURV ${\displaystyle X\sim \operatorname {U} [a,b]}$ is given by the indicator function of its interval of support normalized by the interval's length: $${\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}$$Of particular interest is the uniform distribution on the unit interval ${\displaystyle [0,1]}$. Samples of any desired probability distribution ${\displaystyle \operatorname {D} }$ can be generated by calculating the quantile function of ${\displaystyle \operatorname {D} }$ 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

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).

Real-valued random variables

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

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

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

Example 1

Let ${\displaystyle X}$ be a real-valued, continuous random variable and let ⁠${\displaystyle Y=X^{2}}$⁠.

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

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

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

If ⁠${\displaystyle y\geq 0}$⁠, then

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

so

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

Example 2

Suppose ${\displaystyle X}$ is a random variable with a cumulative distribution

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

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

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

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

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

which is the cumulative distribution function (CDF) of an exponential distribution.

Example 3

Suppose ${\displaystyle X}$ is a random variable with a standard normal distribution, whose density is

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

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

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

In this case the change is not monotonic, because every value of ${\displaystyle Y}$ has two corresponding values of ${\displaystyle X}$ (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,

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

The inverse transformation is

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

and its derivative is

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

Then,

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

This is a chi-squared distribution with one degree of freedom.

Example 4

Suppose ${\displaystyle X}$ is a random variable with a normal distribution, whose density is

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

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

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

In this case the change is not monotonic, because every value of ${\displaystyle Y}$ has two corresponding values of ${\displaystyle X}$ (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:

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

The inverse transformation is

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

and its derivative is

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

Then,

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

This is a noncentral chi-squared distribution with one degree of freedom.

Equality in distribution

If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted ⁠${\displaystyle X~{\stackrel {d}{=}}~Y}$⁠) if they have the same distribution functions:

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

To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.

Almost sure equality

Two random variables X and Y are equal almost surely (denoted ${\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}$) if, and only if, the probability that they are different is zero:

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

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

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

where "ess sup" represents the essential supremum in the sense of measure theory.

Equality

Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space:

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

This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment 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 considered in isolation are "practically equivalent" if they are equal in distribution -- but once we relate them to other 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 variables A, B, C, and D all defined on the same probability space. Suppose that A and B are equal almost surely (${\displaystyle A\;{\stackrel {\text{a.s.}}{=}}\;B}$), but A and C are only equal in distribution (${\displaystyle A~{\stackrel {d}{=}}~C}$). Then ${\displaystyle A+D\;{\stackrel {\text{a.s.}}{=}}\;B+D}$, but in general ${\displaystyle A+D\neq C+D}$ (not even in distribution). Similarly, we have that the expectation values ${\displaystyle \mathbb {E} (AD)=\mathbb {E} (BD)}$, but in general ${\displaystyle \mathbb {E} (AD)\neq \mathbb {E} (CD)}$. Therefore, two random variables that are equal in distribution (but not equal almost surely) can have different covariances with a third random variable.

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