# 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: X, Y, etc.
• Particular realizations of a random variable are written in corresponding lower case letters. For example, x1, x2, …, xn could be a sample corresponding to the random variable 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 φ(z), and its cdf by Φ(z).
• Some common operators:
• 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 x1, x2, ..., xn is often denoted by placing an "overbar" over the symbol, e.g. ${\bar {x}}$ , pronounced "x bar".
• Some commonly used symbols for sample statistics are given below:
• Some commonly used symbols for population parameters are given below:
• the population mean μ,
• the population variance σ2,
• the population standard deviation σ,
• the population correlation ρ,
• the population cumulants κr,
• $x_{(k)}$ 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.

## 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:

• zα or z(α) for the standard normal distribution
• tα,ν or t(α,ν) for the t-distribution with ν degrees of freedom
• ${\chi _{\alpha ,\nu }}^{2}$ or ${\chi }^{2}(\alpha ,\nu )$ for the chi-squared distribution with ν degrees of freedom
• $F_{\alpha ,\nu _{1},\nu _{2}}$ or F(α,ν1,ν2) for the F-distribution with ν1 and ν2 degrees of freedom

## Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. A.
• Column vectors are usually denoted by boldface lowercase letters, e.g. x.
• The transpose operator is denoted by either a superscript T (e.g. AT) or a prime symbol (e.g. A′).
• A row vector is written as the transpose of a column vector, e.g. xT or x′.

## Abbreviations

Common abbreviations include: