# 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, such as X{\textstyle X} or Y{\textstyle 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 $P(X=x)$ 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{\textstyle x}. It is important that X{\textstyle X} and x{\textstyle x} are not confused into meaning the same thing. X{\textstyle X} is an idea, x{\textstyle x} 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, x$_1$, x$_2$, …, x$_n${\textstyle x_{1}, x_{2}, \ldots , x_{n}} could be a sample corresponding to the random variable X{\textstyle X}. A cumulative probability is formally written $P(X\leq x)$ to distinguish the random variable from its realization.
* The probability is sometimes written $\mathbb {P}$ to distinguish it from other functions and measure P 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, $X$ is a random variable that is a function of $\omega$ (i.e., it depends upon $\omega$), and $\omega$ is some outcome of interest within the domain specified by $\Omega$ (say, a particular height, or a particular colour of a car). $\Pr(A)$ 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 φ(z){\textstyle \varphi (z)}, and its cdf by Φ(z){\textstyle \Phi (z)}.
* Some common operators:
   * E[X]{\textstyle \mathrm {E} [X]} : expected value of X
   * var⁡[X]{\textstyle \operatorname {var} [X]} : variance of X
   * cov⁡[X, Y]{\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 $\textstyle B$

## 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 x$_1$, x$_2$, …, x$_n${\textstyle x_{1}, x_{2}, \ldots , x_{n}} is often denoted by placing an "overbar" over the symbol, e.g. ${\bar {x}}$, pronounced " x{\textstyle x} bar".
* Some commonly used symbols for sample statistics are given below:
** the sample mean ${\bar {x}}$,
** the sample variance s$^2${\textstyle s^{2}},
** the sample standard deviation s{\textstyle s},
** the sample correlation coefficient r{\textstyle r},
** the sample cumulants k$_r${\textstyle k_{r}}.
* Some commonly used symbols for population parameters are given below:
** the population mean μ{\textstyle \mu },
** the population variance σ$^2${\textstyle \sigma ^{2}},
** the population standard deviation σ{\textstyle \sigma },
** the population correlation ρ{\textstyle \rho },
** the population cumulants κ$_r${\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 n{\textstyle n}.

## Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability α{\textstyle \alpha }, that is, the value x$_α${\textstyle x_{\alpha }} such that F(x$_α$)=1−α{\textstyle F(x_{\alpha })=1-\alpha }, where F{\textstyle F} is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
* z$_α${\textstyle z_{\alpha }} or z(α){\textstyle z(\alpha )} for the standard normal distribution
* t$_{α, ν}${\textstyle t_{\alpha , \nu }} or t(α, ν){\textstyle t(\alpha , \nu )} for the t-distribution with ν{\textstyle \nu } degrees of freedom
* ${\chi _{\alpha ,\nu }}^{2}$ or ${\chi }^{2}(\alpha ,\nu )$ for the chi-squared distribution with ν{\textstyle \nu } degrees of freedom
* $F_{\alpha ,\nu _{1},\nu _{2}}$ or F(α, ν$_1$, ν$_2$){\textstyle F(\alpha , \nu _{1}, \nu _{2})} for the F-distribution with ν$_1${\textstyle \nu _{1}} and ν$_2${\textstyle \nu _{2}} degrees of freedom

## Linear algebra

* Matrices are usually denoted by boldface capital letters, e.g. A{\textstyle {\mathbf {A}}}.
* Column vectors are usually denoted by boldface lowercase letters, e.g. x{\textstyle {\mathbf {x}}}.
* The transpose operator is denoted by either a superscript T (e.g. A$^T${\textstyle {\mathbf {A}}^{\mathrm {T} }}) or a prime symbol (e.g. A$^′${\textstyle {\mathbf {A}}'}).
* A row vector is written as the transpose of a column vector, e.g. x$^T${\textstyle {\mathbf {x}}^{\mathrm {T} }} or x$^′${\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. $\{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}$