# Notation in probability and statistics

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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 and a cumulative probability is formally written ${\displaystyle P(X>x)}$ to differentiate random variable from realization.
• The probability is sometimes written ${\displaystyle \mathbb {P} }$ to distinguish it from other functions and measure P so as to avoid having to define " P is a probability" and ${\displaystyle \mathbb {P} (A)}$ is short for ${\displaystyle P(\{\omega :X(\omega )\in A\})}$, where ${\displaystyle \omega }$ is an event and ${\displaystyle X(\omega )}$ a corresponding random variable.
• ${\displaystyle \mathbb {P} (A\cap B)}$ or ${\displaystyle \mathbb {P} [A\cap B]}$ indicates the probability that events A and B both occur.
• ${\displaystyle \mathbb {P} (A\cup B)}$ or ${\displaystyle \mathbb {P} [A\cup B]}$ 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 upper case calligraphic (e.g. ${\displaystyle {\mathcal {F}}}$ for the set of sets on which we define the probability P)
• Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
• Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:${\displaystyle {\overline {F}}(x)=1-F(x)}$
• 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 ${\displaystyle X\perp Y}$ or ${\displaystyle X\perp \!\!\!\perp Y}$, and X is independent of Y given W is often written
${\displaystyle X\perp \!\!\!\perp Y\,|\,W}$ or
${\displaystyle X\perp Y\,|\,W}$
• ${\displaystyle \textstyle P(A\mid B)}$, the posterior probability, is the probability of ${\displaystyle \textstyle A}$ given ${\displaystyle \textstyle B}$, i.e., ${\displaystyle \textstyle A}$ after ${\displaystyle \textstyle B}$ 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., ${\displaystyle {\widehat {\theta }}}$ is an estimator for ${\displaystyle \theta }$.
• The arithmetic mean of a series of values x1, x2, ..., xn is often denoted by placing an "overbar" over the symbol, e.g. ${\displaystyle {\bar {x}}}$, 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.

## 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
• ${\displaystyle {\chi _{\alpha ,\nu }}^{2}}$ or ${\displaystyle {\chi }^{2}(\alpha ,\nu )}$ for the chi-squared distribution with ν degrees of freedom
• ${\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}$ or F(α,ν12) 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 lower case 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: