Probability
Outline Catalog of articles Probabilists Glossary Notation Journals Category Mathematics portal
v t e

Statistics
Outline Statisticians Glossary Notation Journals Lists of topics Articles Category Mathematics portal
v t e

Probability theory andstatistics have some commonly used conventions, in addition to standardmathematical notation andmathematical symbols.

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2021) (Learn how and when to remove this message)

• Random variables are usually written inupper case Roman letters, such as$X$ or$Y$ 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${\displaystyle P(X=x)}$ 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),$x$. It is important that$X$ and$x$ are not confused into meaning the same thing.$X$ is an idea,$x$ is a value. Clearly they are related, but they do not have identical meanings.
• Particular realisations of a random variable are written in correspondinglower case letters. For example,$x_1,x_2,\dots ,x_n$ could be asample corresponding to the random variable$X$. A cumulative probability is formally written${\displaystyle P(X\le x)}$ to distinguish the random variable from its realization.^[1]
• The probability is sometimes written${\displaystyle \mathbb{P}}$ to distinguish it from other functions and measureP to avoid having to define "P is a probability" and${\displaystyle \mathbb{P}(X\in A)}$ is short for${\displaystyle P(\{\omega \in \mathrm{\Omega }:X(\omega )\in A\})}$, where${\displaystyle \mathrm{\Omega }}$ is the event space,${\displaystyle X}$ is a random variable that is a function of${\displaystyle \omega }$ (i.e., it depends upon${\displaystyle \omega }$), and${\displaystyle \omega }$ is some outcome of interest within the domain specified by${\displaystyle \mathrm{\Omega }}$ (say, a particular height, or a particular colour of a car).${\displaystyle Pr(A)}$ notation is used alternatively.
• ${\displaystyle \mathbb{P}(A\cap B)}$ or${\displaystyle \mathbb{P}[B\cap A]}$ indicates the probability that eventsA andB both occur. Thejoint probability distribution of random variablesX andY is denoted as${\displaystyle P(X,Y)}$, while joint probability mass function or probability density function as${\displaystyle f(x,y)}$ and joint cumulative distribution function as${\displaystyle F(x,y)}$.
• ${\displaystyle \mathbb{P}(A\cup B)}$ or${\displaystyle \mathbb{P}[B\cup A]}$ indicates the probability of either eventA or eventB occurring ("or" in this case meansone or the other or both).
• σ-algebras are usually written with uppercasecalligraphic (e.g.${\displaystyle \mathcal{F}}$ for the set of sets on which we define the probabilityP)
• Probability density functions (pdfs) andprobability mass functions are denoted by lowercase letters, e.g.${\displaystyle f(x)}$, or${\displaystyle f_X(x)}$.
• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g.${\displaystyle F(x)}$, or${\displaystyle F_X(x)}$.
• Survival functions or complementary cumulative distribution functions are often denoted by placing anoverbar over the symbol for the cumulative:${\displaystyle \overline{F}(x)=1-F(x)}$, or denoted as${\displaystyle S(x)}$,
• In particular, the pdf of thestandard normal distribution is denoted by$\phi (z)$, and its cdf by$\mathrm{\Phi }(z)$.
• Some common operators:

• $\mathrm{E}[X]$:expected value ofX
• $\mathrm{var}[X]$:variance ofX
• $\mathrm{cov}[X,Y]$:covariance ofX andY

• X is independent of Y is often written${\displaystyle X\perp Y}$ or${\displaystyle X\perp \perp Y}$, and X is independent of Y given W is often written

${\displaystyle X\perp \perp Y|W}$ or

${\displaystyle X\perp Y|W}$

• ${\displaystyle P(A\mid B)}$, theconditional probability, is the probability of${\displaystyle A}$given${\displaystyle B}$^[2]

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2021) (Learn how and when to remove this message)

• Greek letters (e.g.θ,β) are commonly used to denote unknown parameters (population parameters).^[3]
• A tilde (~) denotes "has the probability distribution of".
• Placing a hat, or caret (also known as a circumflex), over a true parameter denotes anestimator of it, e.g.,${\displaystyle \hat{\theta }}$ is an estimator for${\displaystyle \theta }$.
• Thearithmetic mean of a series of values$x_1,x_2,\dots ,x_n$ is often denoted by placing an "overbar" over the symbol, e.g.${\displaystyle \overline{x}}$, pronounced "$x$ bar".
• Some commonly used symbols forsample statistics are given below:
  • thesample mean${\displaystyle \overline{x}}$,
  • thesample variance$s^2$,
  • thesample standard deviation$s$,
  • thesample correlation coefficient$r$,
  • the sample cumulants$k_r$.
• Some commonly used symbols forpopulation parameters are given below:
  • the population mean$\mu$,
  • the population variance${\sigma }^2$,
  • the population standard deviation$\sigma$,
  • the populationcorrelation$\rho$,
  • the populationcumulants${\kappa }_r$,
• ${\displaystyle x_{(k)}}$ is used for the${\displaystyle k^{\text{th}}}$order statistic, where${\displaystyle x_{(1)}}$ is the sample minimum and${\displaystyle x_{(n)}}$ is the sample maximum from a total sample size$n$.^[4]

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2021) (Learn how and when to remove this message)

Theα-level uppercritical value of aprobability distribution is the value exceeded with probability$\alpha$, that is, the value$x_{\alpha }$ such that$F(x_{\alpha })=1-\alpha$, where$F$ is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

• $z_{\alpha }$ or$z(\alpha )$ for thestandard normal distribution
• $t_{\alpha ,\nu }$ or$t(\alpha ,\nu )$ for thet-distribution with$\nu$degrees of freedom
• ${\displaystyle {{\chi }_{\alpha ,\nu }}^2}$ or${\displaystyle {\chi }^2(\alpha ,\nu )}$ for thechi-squared distribution with$\nu$ degrees of freedom
• ${\displaystyle F_{\alpha ,{\nu }_1,{\nu }_2}}$ or$F(\alpha ,{\nu }_1,{\nu }_2)$ for theF-distribution with${\nu }_1$ and${\nu }_2$ degrees of freedom

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2021) (Learn how and when to remove this message)

• Matrices are usually denoted by boldface capital letters, e.g.$\mathbf{A}$.
• Column vectors are usually denoted by boldface lowercase letters, e.g.$\mathbf{x}$.
• Thetranspose operator is denoted by either a superscript T (e.g.${\mathbf{A}}^{\mathrm{T}}$) or aprime symbol (e.g.${\mathbf{A}}^{\prime }$).
• Arow vector is written as the transpose of a column vector, e.g.${\mathbf{x}}^{\mathrm{T}}$ or${\mathbf{x}}^{\prime }$.

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(March 2021) (Learn how and when to remove this message)

Common abbreviations include:

• a.e.almost everywhere
• a.s.almost surely
• cdfcumulative distribution function
• cmfcumulative mass function
• dfdegrees of freedom (also${\displaystyle \nu }$)
• i.i.d.independent and identically distributed
• pdfprobability density function
• pmfprobability mass function
• r.v.random variable
• w.p. with probability;wp1with probability 1
• i.o. infinitely often, i.e.${\displaystyle \{A_n\text{ i.o.}\}=\bigcap _N\bigcup _{n\ge N}{A}_n}$
• ult. ultimately, i.e.${\displaystyle \{A_n\text{ ult.}\}=\bigcup _N\bigcap _{n\ge N}{A}_n}$

• Glossary of probability and statistics
• Combinations andpermutations
• History of mathematical notation

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

• Probability and statistics
• Mathematical notation

• Articles with short description
• Short description with empty Wikidata description
• Articles needing additional references from March 2021
• All articles needing additional references
• Pages that use a deprecated format of the math tags