# 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, 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 α, that is, the value xα such that F(xα) = 1 − α where 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: