Random variable

 ( Statistical Randomness ) 

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 = the value of the spinner as in the preceding example. There is a probability of 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 . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).

The measure-theoretic definition is as follows.

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

In more intuitive terms, a member of $\backslash Omega$ is a possible outcome, a member of $\backslash mathcal\{F\}$ 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 $\backslash mathcal\{E\}$ 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 a topological space, then the most common choice for the σ-algebra $\backslash mathcal\{E\}$ is the Borel σ-algebra $\backslash mathcal\{B\}(E)$, which is the σ-algebra generated by the collection of all open sets in $E$. In such case the $(E,\; \backslash mathcal\{E\})$-valued random variable is called an $E$-valued random variable. Moreover, when the space $E$ is the real line $\backslash mathbb\{R\}$, then such a real-valued random variable is called simply a random variable.

$\backslash \{\; \backslash omega\; :\; X(\backslash omega)\; \backslash le\; r\; \backslash \}\; \backslash in\; \backslash mathcal\{F\}\; \backslash qquad\; \backslash forall\; r\; \backslash in\; \backslash mathbb\{R\}.$

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

Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables $X$, find a collection $\backslash \{f\_i\backslash \}$ of functions such that the expectation values $\backslash operatorname\{E\}f\_i(X)$ fully characterise the distribution of 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 a categorical random variable X that can take on the nominal data values "red", "blue" or "green", the real-valued function $X$ can be constructed; this uses the Iverson bracket, and has the value 1 if $X$ has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.

$F\_Y(y)\; =\; \backslash operatorname\{P\}(g(X)\; \backslash le\; y).$

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

$F\_Y(y)\; =\; \backslash operatorname\{P\}(g(X)\; \backslash le\; y)\; =$

& \text{if } h = g^{-1} \text{ increasing} ,\\
     

& \text{if } h = g^{-1} \text{ decreasing} .
     

With the same hypotheses of invertibility of $g$, assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to $y$, in order to obtain

$f\_Y(y)\; =\; f\_X\backslash bigl(h(y)\backslash bigr)\; \backslash left|\; \backslash frac\{d\; h(y)\}\{d\; y\}\; \backslash right|.$

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 the probability density functions can be generalized with

$f\_Y(y)\; =\; \backslash sum\_\{i\}\; f\_X(g\_\{i\}^\{-1\}(y))\; \backslash left|\; \backslash frac\{d\; g\_\{i\}^\{-1\}(y)\}\{d\; y\}\; \backslash right|$

where $x\_i\; =\; g\_i^\{-1\}(y)$, according to the inverse function theorem. The formulas for densities do not demand $g$ to be increasing.

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

$F\_Y(y)\; =\; \backslash operatorname\{P\}(X^2\; \backslash le\; y).$

If , then $P(X^2\; \backslash leq\; y)\; =\; 0$, so

If $y\; \backslash geq\; 0$, then

$\backslash operatorname\{P\}(X^2\; \backslash le\; y)\; =\; \backslash operatorname\{P\}(|X|\; \backslash le\; \backslash sqrt\{y\})$

= \operatorname{P}(-\sqrt{y} \le  X \le \sqrt{y}),
     

so

$F\_Y(y)\; =\; F\_X(\backslash sqrt\{y\})\; -\; F\_X(-\backslash sqrt\{y\})\backslash qquad\backslash hbox\{if\}\backslash quad\; y\; \backslash ge\; 0.$

$F\_\{X\}(x)\; =\; P(X\; \backslash leq\; x)\; =\; \backslash frac\{1\}\{(1\; +\; e^\{-x\})^\{\backslash theta\}\}$

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

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

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

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

$f\_X(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}e^\{-x^2/2\}.$

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

$f\_Y(y)\; =\; \backslash sum\_\{i\}\; f\_X(g\_\{i\}^\{-1\}(y))\; \backslash left|\; \backslash frac\{d\; g\_\{i\}^\{-1\}(y)\}\{d\; y\}\; \backslash right|.$

In this case the change is not monotonic, 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.,

$f\_Y(y)\; =\; 2f\_X(g^\{-1\}(y))\; \backslash left|\; \backslash frac\{d\; g^\{-1\}(y)\}\{d\; y\}\; \backslash right|.$

The inverse transformation is

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

$\backslash frac\{d\; g^\{-1\}(y)\}\{d\; y\}\; =\; \backslash frac\{1\}\{2\backslash sqrt\{y\}\}\; .$

Then,

$f\_Y(y)\; =\; 2\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}e^\{-y/2\}\; \backslash frac\{1\}\{2\backslash sqrt\{y\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\; y\}\}e^\{-y/2\}.$

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

$f\_X(x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash sigma^2\}\}e^\{-(x-\backslash mu)^2/(2\backslash sigma^2)\}.$

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

$f\_Y(y)\; =\; \backslash sum\_\{i\}\; f\_X(g\_\{i\}^\{-1\}(y))\; \backslash left|\; \backslash frac\{d\; g\_\{i\}^\{-1\}(y)\}\{d\; y\}\; \backslash right|.$

In this case the change is not monotonic, 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:

$f\_Y(y)\; =\; f\_X(g\_1^\{-1\}(y))\backslash left|\backslash frac\{d\; g\_1^\{-1\}(y)\}\{d\; y\}\; \backslash right|\; +f\_X(g\_2^\{-1\}(y))\backslash left|\; \backslash frac\{d\; g\_2^\{-1\}(y)\}\{d\; y\}\; \backslash right|.$

The inverse transformation is

$x\; =\; g\_\{1,2\}^\{-1\}(y)\; =\; \backslash pm\; \backslash sqrt\{y\}$

$\backslash frac\{d\; g\_\{1,2\}^\{-1\}(y)\}\{d\; y\}\; =\; \backslash pm\; \backslash frac\{1\}\{2\backslash sqrt\{y\}\}\; .$

Then,

$f\_Y(y)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash sigma^2\}\}\; \backslash frac\{1\}\{2\backslash sqrt\{y\}\}\; (e^\{-(\backslash sqrt\{y\}-\backslash mu)^2/(2\backslash sigma^2)\}+e^\{-(-\backslash sqrt\{y\}-\backslash mu)^2/(2\backslash sigma^2)\})\; .$

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

• The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
• Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).

In increasing order of strength, the precise definition of these notions of equivalence is given below.

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

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.

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

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

$d\_\backslash infty(X,Y)=\backslash operatorname\{ess\}\; \backslash sup\_\backslash omega|X(\backslash omega)-Y(\backslash omega)|,$

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

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

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.

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 ($A\; \backslash ;\; \backslash stackrel\{\backslash text\{a.s.\}\}\{=\}\; \backslash ;\; B$), but A and C are only equal in distribution ($A\; \backslash stackrel\{d\}\{=\}\; C$). Then $A\; +\; D\; \backslash ;\; \backslash stackrel\{\backslash text\{a.s.\}\}\{=\}\; \backslash ;\; B\; +\; D$, but in general $A\; +\; D\; \backslash ;\; \backslash neq\; \backslash ;\; C\; +\; D$ (not even in distribution). Similarly, we have that the expectation values $\backslash mathbb\{E\}(AD)\; =\; \backslash mathbb\{E\}(BD)$, but in general $\backslash mathbb\{E\}(AD)\; \backslash neq\; \backslash mathbb\{E\}(CD)$. Therefore, two random variables that are equal in distribution (but not equal almost surely) can have different covariance with a third random variable.

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 on convergence of random variables.

• 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

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• Olav Kallenberg (2025). 9780387953137, Springer Verlag. . ISBN 9780387953137
• Athanasios Papoulis (1965). 9780071199810, McGraw–Hill. . ISBN 9780071199810