View source for 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 [[Table of mathematical symbols|mathematical symbols]].

==Probability theory==
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* [[Random variable]]s are usually written in [[upper case]] Roman letters, such as <math display="inline">X</math> or <math display="inline">Y</math> and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if <math>P(X = x) </math> is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), ''X'', would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), <math display="inline">x</math>. It is important that <math display="inline">X</math> and <math display="inline">x</math> are not confused into meaning the same thing. <math display="inline">X</math> is an idea, <math display="inline">x</math> is a value. Clearly they are related, but they do not have identical meanings.
* Particular realisations of a random variable are written in corresponding [[lower case]] letters. For example, <math display="inline">x_1,x_2, \ldots,x_n</math> could be a [[random sample|sample]] corresponding to the random variable                                                                                                                                                                                            <math display="inline">X</math>. A cumulative probability is formally written <math>P(X\le x) </math> to distinguish the random variable from its realization.<ref>{{Cite web |date=2021-08-09 |title=Calculating Probabilities from Cumulative Distribution Function |url=https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/ |access-date=2024-02-26}}</ref>
* The probability is sometimes written <math>\mathbb{P} </math> to distinguish it from other functions and measure ''P'' to avoid having to define "''P'' is a probability" and <math>\mathbb{P}(X\in A) </math> is short for <math>P(\{\omega \in\Omega: X(\omega) \in A\})</math>, where <math>\Omega</math> is the event space, <math>X</math> is a random variable that is a function of <math>\omega</math> (i.e., it depends upon <math>\omega</math>), and <math>\omega</math> is some outcome of interest within the domain specified by <math>\Omega</math> (say, a particular height, or a particular colour of a car). <math>\Pr(A)</math> notation is used alternatively.
*<math>\mathbb{P}(A \cap B)</math> or <math>\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>P(X, Y)</math>, while joint probability mass function or probability density function as <math>f(x, y)</math> and joint cumulative distribution function as <math>F(x, y)</math>.
*<math>\mathbb{P}(A \cup B)</math> or <math>\mathbb{P}[B \cup A]</math> indicates the probability of either event ''A'' or event ''B'' occurring ("or" in this case means [[inclusive or|one or the other or both]]).
*[[sigma-algebra|σ-algebras]] are usually written with uppercase [[Calligraphy|calligraphic]] (e.g. <math>\mathcal F</math> for the set of sets on which we define the probability ''P'')
*[[Probability density function]]s (pdfs) and [[probability mass function]]s are denoted by lowercase letters, e.g. <math>f(x)</math>, or <math>f_X(x)</math>.
*[[Cumulative distribution function]]s (cdfs) are denoted by uppercase letters, e.g. <math>F(x)</math>, or <math>F_X(x)</math>.
* [[Survival function]]s or complementary cumulative distribution functions are often denoted by placing an [[overbar]] over the symbol for the cumulative:<math>\overline{F}(x) =1-F(x)</math>, or denoted as <math>S(x)</math>,
*In particular, the pdf of the [[standard normal distribution]] is denoted by <math display="inline">\varphi(z)</math>, and its cdf by <math display="inline">\Phi(z)</math>.
*Some common operators: 
:* <math display="inline">\mathrm{E}[X]</math> : [[expected value]] of ''X''
:* <math display="inline">\operatorname{var}[X]</math> : [[variance]] of ''X''
:* <math display="inline">\operatorname{cov}[X,Y]</math> : [[covariance]] of ''X'' and ''Y''
* X is independent of Y is often written <math>X \perp Y</math> or <math>X \perp\!\!\!\perp Y</math>, and X is independent of Y given W is often written 
:<math>X \perp\!\!\!\perp Y \,|\, W </math> or
:<math>X \perp Y \,|\, W</math>
* <math>\textstyle P(A\mid B)</math>, the ''[[conditional probability]]'', is the probability of <math>\textstyle A</math> ''given'' <math>\textstyle B</math> <ref>{{Citation |title=Probability and stochastic processes |date=2013-07-22 |url=http://dx.doi.org/10.1201/b15257-3 |work=Applied Stochastic Processes |pages=9–36 |access-date=2023-12-08 |publisher=Chapman and Hall/CRC |doi=10.1201/b15257-3 |isbn=978-0-429-16812-3}}</ref>

==Statistics==
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*Greek letters (e.g. ''&theta;'', ''&beta;'') are commonly used to denote unknown parameters (population parameters).<ref>{{Cite web |date=1999-02-13 |title=Letters of the Greek Alphabet and Some of Their Statistical Uses |url=https://lesn.appstate.edu/olson/EDL7150/Components/Other%20useful%20links/Greek%20Alphabet%20and%20Statistics.htm |access-date=2024-02-26 |website=les.appstate.edu/}}</ref>
*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>\widehat{\theta}</math> is an estimator for <math>\theta</math>.
*The [[arithmetic mean]] of a series of values <math display="inline">x_1,x_2, \ldots,x_n</math> is often denoted by placing an "[[overbar]]" over the symbol, e.g. <math>\bar{x}</math>, pronounced "<math display="inline">x</math> bar".
*Some commonly used symbols for [[Sample (statistics)|sample]] statistics are given below:
**the [[sample mean]] <math>\bar{x}</math>,
**the [[sample variance]] <math display="inline">s^2</math>,
** the [[sample standard deviation]] ''<math display="inline">s</math>'',
**the [[Pearson correlation coefficient|sample correlation coefficient]] ''<math display="inline">r</math>'',
**the sample cumulants <math display="inline">k_r</math>.
*Some commonly used symbols for [[Statistical population|population]] parameters are given below:
**the population mean <math display="inline">\mu</math>,
**the population variance <math display="inline">\sigma^2</math>,
** the population standard deviation ''<math display="inline">\sigma</math>'',
**the population [[Pearson product-moment correlation coefficient|correlation]] ''<math display="inline">\rho</math>'',
**the population [[cumulant]]s ''<math display="inline">\kappa_r</math>'',
*<math>x_{(k)}</math> is used for the <math>k^\text{th}</math> [[order statistic]], where <math>x_{(1)}</math> is the sample minimum and <math>x_{(n)}</math> is the sample maximum from a total sample size <math display="inline">n</math>.<ref>{{Cite web |title=Order Statistics |url=https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf |access-date=2024-02-26 |website=colorado.edu}}</ref>

==Critical values==
{{Unreferenced section|date=March 2021}}
The ''α''-level upper [[critical value (statistics)|critical value]] of a [[probability distribution]] is the value exceeded with probability <math display="inline">\alpha</math>, that is, the value <math display="inline">x_\alpha</math> such that <math display="inline">F(x_\alpha) = 1-\alpha</math>, where <math display="inline">F</math> is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
*<math display="inline">z_\alpha</math> or <math display="inline">z(\alpha)</math> for the [[standard normal distribution]]
*<math display="inline">t_{\alpha,\nu}</math> or <math display="inline">t(\alpha,\nu)</math> for the [[Student's t-distribution|''t''-distribution]] with <math display="inline">\nu</math> [[Degrees of freedom (statistics)|degrees of freedom]]
*<math>{\chi_{\alpha,\nu}}^2</math> or <math>{\chi}^{2}(\alpha,\nu)</math> for the [[chi-squared distribution]] with <math display="inline">\nu</math> degrees of freedom
*<math>F_{\alpha,\nu_1,\nu_2}</math> or <math display="inline">F(\alpha,\nu_1,\nu_2)</math> for the [[F-distribution]] with <math display="inline">\nu_1</math> and <math display="inline">\nu_2</math> degrees of freedom

==Linear algebra==
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*[[Matrix (mathematics)|Matrices]] are usually denoted by boldface capital letters, e.g. <math display="inline">\bold{A}</math>. 
*[[Column vector]]s are usually denoted by boldface lowercase letters, e.g. '''<math display="inline">\bold{x}</math>'''.
*The [[transpose]] operator is denoted by either a superscript T (e.g. '''<math display="inline">\bold{A}^\mathrm{T}</math>''') or a [[prime (symbol)|prime symbol]] (e.g. '''<math display="inline">\bold{A}'</math>''').
*A [[row vector]] is written as the transpose of a column vector, e.g. '''<math display="inline">\bold{x}^\mathrm{T}</math>''' or '''<math display="inline">\bold{x}'</math>'''.

==Abbreviations==
{{Unreferenced section|date=March 2021}}
Common abbreviations include:
*'''a.e.''' [[almost everywhere]]
*'''a.s.''' [[almost surely]]
* '''cdf''' [[cumulative distribution function]]
* '''cmf''' [[cumulative mass function]]
*'''df''' [[degrees of freedom (statistics)|degrees of freedom]] (also <math>\nu</math>)
*'''i.i.d.''' [[Independent and identically distributed random variables|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> \{ A_n\text{ i.o.} \} = \bigcap_N\bigcup_{n\geq N} A_n </math>
* '''ult.''' ultimately, i.e. <math>\{ A_n \text{ ult.} \} = \bigcup_N\bigcap_{n\geq N} A_n </math>

== See also ==
*[[Glossary of probability and statistics]]
*[[Combination]]s and [[permutation]]s
*[[History of mathematical notation]]

== References ==
{{Reflist}}
{{refbegin}}
* {{Citation |title = Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation |first1 = Max |last1 = Halperin |first2 = H. O. |last2 = Hartley |first3 = P. G. |last3 = Hoel |journal = The American Statistician |volume = 19 |year = 1965 |pages = 12–14 |issue = 3 |doi = 10.2307/2681417 |jstor = 2681417 }}
{{refend}}

== External links ==
* [http://jeff560.tripod.com/stat.html Earliest Uses of Symbols in Probability and Statistics], maintained by Jeff Miller.

{{Mathematical symbols notation language}}

[[Category:Probability and statistics| Notation]]
[[Category:Mathematical notation]]