Notation in probability and statistics

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

Probability theory

Random variables are usually written in upper case roman letters: ${\textstyle X}$ , ${\textstyle Y}$ , etc.
Particular realizations of a random variable are written in corresponding lower case letters. For example, ${\textstyle x_{1},x_{2},\ldots ,x_{n}}$ could be a sample corresponding to the random variable ${\textstyle X}$ . A cumulative probability is formally written $P(X\leq x)$ to differentiate the random variable from its realization.
The probability is sometimes written $\mathbb {P}$ to distinguish it from other functions and measure P so as to avoid having to define "P is a probability" and $\mathbb {P} (X\in A)$ is short for $P(\{\omega \in \Omega :X(\omega )\in A\})$ , where $\Omega$ is the event space and $X(\omega )$ is a random variable. $\Pr(A)$ notation is used alternatively.
$\mathbb {P} (A\cap B)$ or $\mathbb {P} [B\cap A]$ indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as $P(X,Y)$ , while joint probability mass function or probability density function as $f(x,y)$ and joint cumulative distribution function as $F(x,y)$ .
$\mathbb {P} (A\cup B)$ or $\mathbb {P} [B\cup A]$ 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. ${\mathcal {F}}$ 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. $f(x)$ , or $f_{X}(x)$ .
Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. $F(x)$ , or $F_{X}(x)$ .
Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative: ${\overline {F}}(x)=1-F(x)$ , or denoted as $S(x)$ ,
In particular, the pdf of the standard normal distribution is denoted by ${\textstyle \varphi (z)}$ , and its cdf by ${\textstyle \Phi (z)}$ .
Some common operators:

${\textstyle \mathrm {E} [X]}$ : expected value of X
${\textstyle \operatorname {var} [X]}$ : variance of X
${\textstyle \operatorname {cov} [X,Y]}$ : covariance of X and Y

X is independent of Y is often written $X\perp Y$ or $X\perp \!\!\!\perp Y$ , and X is independent of Y given W is often written

X\perp \!\!\!\perp Y\,|\,W

or

X\perp Y\,|\,W

$\textstyle P(A\mid B)$ , the conditional probability, is the probability of $\textstyle A$ given $\textstyle B$ , i.e., $\textstyle A$ after $\textstyle B$ is observed.^{[citation needed]}

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., ${\widehat {\theta }}$ is an estimator for $\theta$ .
The arithmetic mean of a series of values ${\textstyle x_{1},x_{2},\ldots ,x_{n}}$ is often denoted by placing an "overbar" over the symbol, e.g. ${\bar {x}}$ , pronounced " ${\textstyle x}$ bar".
Some commonly used symbols for sample statistics are given below:
- the sample mean ${\bar {x}}$ ,
- the sample variance ${\textstyle s^{2}}$ ,
- the sample standard deviation ${\textstyle s}$ ,
- the sample correlation coefficient ${\textstyle r}$ ,
- the sample cumulants ${\textstyle k_{r}}$ .
Some commonly used symbols for population parameters are given below:
- the population mean ${\textstyle \mu }$ ,
- the population variance ${\textstyle \sigma ^{2}}$ ,
- the population standard deviation ${\textstyle \sigma }$ ,
- the population correlation ${\textstyle \rho }$ ,
- the population cumulants ${\textstyle \kappa _{r}}$ ,
$x_{(k)}$ is used for the $k^{\text{th}}$ order statistic, where $x_{(1)}$ is the sample minimum and $x_{(n)}$ is the sample maximum from a total sample size ${\textstyle n}$ .

Critical values

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

${\textstyle z_{\alpha }}$ or ${\textstyle z(\alpha )}$ for the standard normal distribution
${\textstyle t_{\alpha ,\nu }}$ or ${\textstyle t(\alpha ,\nu )}$ for the t-distribution with ${\textstyle \nu }$ degrees of freedom
${\chi _{\alpha ,\nu }}^{2}$ or ${\chi }^{2}(\alpha ,\nu )$ for the chi-squared distribution with ${\textstyle \nu }$ degrees of freedom
$F_{\alpha ,\nu _{1},\nu _{2}}$ or ${\textstyle F(\alpha ,\nu _{1},\nu _{2})}$ for the F-distribution with ${\textstyle \nu _{1}}$ and ${\textstyle \nu _{2}}$ degrees of freedom

Linear algebra

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

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 $\nu$ )
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. $\{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}$
ult. ultimately, i.e. $\{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}$

References

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, JSTOR 2681417

External links

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