Notation in probability and statistics

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Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

## Contents

* 1 Probability theory
* 2 Statistics
* 3 Critical values
* 4 Linear algebra
* 5 Abbreviations
* 6 See also
* 7 References
* 8 External links

## Probability theory

* Random variables are usually written in upper case roman letters: [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], etc.
* Particular realizations of a random variable are written in corresponding lower case letters. For example, [math]\displaystyle{ x_1,x_2, \ldots,x_n }[/math] could be a sample corresponding to the random variable [math]\displaystyle{ X }[/math]. A cumulative probability is formally written [math]\displaystyle{ P(X\le x) }[/math] to differentiate the random variable from its realization.
* The probability is sometimes written [math]\displaystyle{ \mathbb{P} }[/math] to distinguish it from other functions and measure P so as to avoid having to define "P is a probability" and [math]\displaystyle{ \mathbb{P}(X\in A) }[/math] is short for [math]\displaystyle{ P(\\{\omega \in\Omega: X(\omega) \in A\\}) }[/math], where [math]\displaystyle{ \Omega }[/math] is the event space and [math]\displaystyle{ X(\omega) }[/math] is a random variable. [math]\displaystyle{ \Pr(A) }[/math] notation is used alternatively.
* [math]\displaystyle{ \mathbb{P}(A \cap B) }[/math] or [math]\displaystyle{ \mathbb{P}[B \cap A] }[/math] indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as [math]\displaystyle{ P(X, Y) }[/math], while joint probability mass function or probability density function as [math]\displaystyle{ f(x, y) }[/math] and joint cumulative distribution function as [math]\displaystyle{ F(x, y) }[/math].
* [math]\displaystyle{ \mathbb{P}(A \cup B) }[/math] or [math]\displaystyle{ \mathbb{P}[B \cup A] }[/math] indicates the probability of either event A or event B occurring ("or" in this case means one or the other or both).
* σ-algebras are usually written with uppercase calligraphic (e.g. [math]\displaystyle{ \mathcal F }[/math] for the set of sets on which we define the probability P)
* Probability density functions (pdfs) and probability mass functions are denoted by lowercase letters, e.g. [math]\displaystyle{ f(x) }[/math], or [math]\displaystyle{ f_X(x) }[/math].
* Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. [math]\displaystyle{ F(x) }[/math], or [math]\displaystyle{ F_X(x) }[/math].
* Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:[math]\displaystyle{ \overline{F}(x) =1-F(x) }[/math], or denoted as [math]\displaystyle{ S(x) }[/math],
* In particular, the pdf of the standard normal distribution is denoted by [math]\displaystyle{ \varphi(z) }[/math], and its cdf by [math]\displaystyle{ \Phi(z) }[/math].
* Some common operators:

* [math]\displaystyle{ \mathrm{E}[X] }[/math] : expected value of X
* [math]\displaystyle{ \operatorname{var}[X] }[/math] : variance of X
* [math]\displaystyle{ \operatorname{cov}[X,Y] }[/math] : covariance of X and Y

* X is independent of Y is often written [math]\displaystyle{ X \perp Y }[/math] or [math]\displaystyle{ X \perp\\!\\!\\!\perp Y }[/math], and X is independent of Y given W is often written

[math]\displaystyle{ X \perp\\!\\!\\!\perp Y \,|\, W }[/math] or
[math]\displaystyle{ X \perp Y \,|\, W }[/math]

* [math]\displaystyle{ \textstyle P(A\mid B) }[/math], the conditional probability, is the probability of [math]\displaystyle{ \textstyle A }[/math] given [math]\displaystyle{ \textstyle B }[/math] [1]

## Statistics

* Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
* A tilde (~) denotes "has the probability distribution of".
* Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an estimator of it, e.g., [math]\displaystyle{ \widehat{\theta} }[/math] is an estimator for [math]\displaystyle{ \theta }[/math].
* The arithmetic mean of a series of values [math]\displaystyle{ x_1,x_2, \ldots,x_n }[/math] is often denoted by placing an "overbar" over the symbol, e.g. [math]\displaystyle{ \bar{x} }[/math], pronounced "[math]\displaystyle{ x }[/math] bar".
* Some commonly used symbols for sample statistics are given below:
* the sample mean [math]\displaystyle{ \bar{x} }[/math],
* the sample variance [math]\displaystyle{ s^2 }[/math],
* the sample standard deviation [math]\displaystyle{ s }[/math],
* the sample correlation coefficient [math]\displaystyle{ r }[/math],
* the sample cumulants [math]\displaystyle{ k_r }[/math].
* Some commonly used symbols for population parameters are given below:
* the population mean [math]\displaystyle{ \mu }[/math],
* the population variance [math]\displaystyle{ \sigma^2 }[/math],
* the population standard deviation [math]\displaystyle{ \sigma }[/math],
* the population correlation [math]\displaystyle{ \rho }[/math],
* the population cumulants [math]\displaystyle{ \kappa_r }[/math],
* [math]\displaystyle{ x_{(k)} }[/math] is used for the [math]\displaystyle{ k^\text{th} }[/math] order statistic, where [math]\displaystyle{ x_{(1)} }[/math] is the sample minimum and [math]\displaystyle{ x_{(n)} }[/math] is the sample maximum from a total sample size [math]\displaystyle{ n }[/math].

## Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability [math]\displaystyle{ \alpha }[/math], that is, the value [math]\displaystyle{ x_\alpha }[/math] such that [math]\displaystyle{ F(x_\alpha) = 1-\alpha }[/math], where [math]\displaystyle{ F }[/math] is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

* [math]\displaystyle{ z_\alpha }[/math] or [math]\displaystyle{ z(\alpha) }[/math] for the standard normal distribution
* [math]\displaystyle{ t_{\alpha,\nu} }[/math] or [math]\displaystyle{ t(\alpha,\nu) }[/math] for the t-distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom
* [math]\displaystyle{ {\chi_{\alpha,\nu}}^2 }[/math] or [math]\displaystyle{ {\chi}^{2}(\alpha,\nu) }[/math] for the chi-squared distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom
* [math]\displaystyle{ F_{\alpha,\nu_1,\nu_2} }[/math] or [math]\displaystyle{ F(\alpha,\nu_1,\nu_2) }[/math] for the F-distribution with [math]\displaystyle{ \nu_1 }[/math] and [math]\displaystyle{ \nu_2 }[/math] degrees of freedom

## Linear algebra

* Matrices are usually denoted by boldface capital letters, e.g. [math]\displaystyle{ \bold{A} }[/math].
* Column vectors are usually denoted by boldface lowercase letters, e.g. [math]\displaystyle{ \bold{x} }[/math].
* The transpose operator is denoted by either a superscript T (e.g. [math]\displaystyle{ \bold{A}^\mathrm{T} }[/math]) or a prime symbol (e.g. [math]\displaystyle{ \bold{A}' }[/math]).
* A row vector is written as the transpose of a column vector, e.g. [math]\displaystyle{ \bold{x}^\mathrm{T} }[/math] or [math]\displaystyle{ \bold{x}' }[/math].

## Abbreviations

Common abbreviations include:

* a.e. almost everywhere
* a.s. almost surely
* cdf cumulative distribution function
* cmf cumulative mass function
* df degrees of freedom (also [math]\displaystyle{ \nu }[/math])
* i.i.d. independent and identically distributed
* pdf probability density function
* pmf probability mass function
* r.v. random variable
* w.p. with probability; wp1 with probability 1
* i.o. infinitely often, i.e. [math]\displaystyle{ \\{ A_n\text{ i.o.} \\} = \bigcap_N\bigcup_{n\geq N} A_n }[/math]
* ult. ultimately, i.e. [math]\displaystyle{ \\{ A_n \text{ ult.} \\} = \bigcup_N\bigcap_{n\geq N} A_n }[/math]

## See also

* Glossary of probability and statistics
* Combinations and permutations
* History of mathematical notation

## References

1. ↑ "Probability and stochastic processes", Applied Stochastic Processes (Chapman and Hall/CRC): pp. 9–36, 2013-07-22, ISBN 978-0-429-16812-3, http://dx.doi.org/10.1201/b15257-3, retrieved 2023-12-08

* Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", The American Statistician 19 (3): 12–14, doi:10.2307/2681417

## External links

* Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.

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