Notation in probability

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, ''x''₁, ''x''₂, …, ''x''_''n'' could be a sample corresponding to the random variable ''X''. A cumulative probability is formally written $P(X\le x)$ to differentiate the random variable from its realization.
* The probability is sometimes written $\mathbb$ to distinguish it from other functions and measure ''P'' so as to avoid having to define "''P'' is a probability" and $\mathbb(X\in A)$ is short for $P(\)$, where $\Omega$ is the event space and $X(\omega)$ is a random variable. $\Pr(A)$ notation is used alternatively.
*$\mathbb(A \cap B)$ or \mathbb \cap A/math> 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 $F(x, y)$.
*$\mathbb(A \cup B)$ or \mathbb \cup A/math> 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:$\overline(x) =1-F(x)$, 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:
:* E 'X'': expected value of ''X''
:* var 'X'': variance of ''X''
:* 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.

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., $\widehat$ is an estimator for $\theta$.
*The arithmetic mean of a series of values ''x''₁, ''x''₂, ..., ''x''_''n'' is often denoted by placing an "overbar" over the symbol, e.g. $\bar$, pronounced "''x'' bar".
*Some commonly used symbols for sample statistics are given below:
**the sample mean $\bar$,
**the sample variance ''s''²,
** the sample standard deviation ''s'',
**the sample correlation coefficient ''r'',
**the sample cumulants ''k_r''.
*Some commonly used symbols for population parameters are given below:
**the population mean ''μ'',
**the population variance ''σ''²,
** the population standard deviation ''σ'',
**the population correlation ''ρ'',
**the population cumulants ''κ_r'',
*$x_$ is used for the $k^\text$ order statistic, where $x_$ is the sample minimum and $x_$ 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 ''x''_''α'' such that ''F''(''x''_''α'') = 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:
*''z''_''α'' or ''z''(''α'') for the standard normal distribution
*''t''_{''α'',''ν''} or ''t''(''α'',''ν'') for the ''t''-distribution with ν degrees of freedom
*$^2$ or $^(\alpha,\nu)$ for the chi-squared distribution with ν degrees of freedom
*$F_$ or F(α,''ν''₁,''ν''₂) for the F-distribution with ''ν''₁ and ''ν''₂ 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. A^T) or a prime symbol (e.g. A′).
*A row vector is written as the transpose of a column vector, e.g. x^T or 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. $\ = \bigcap_N\bigcup_ A_n$
* ult. ultimately, i.e. $\ = \bigcup_N\bigcap_ A_n$

See also

*Glossary of probability and statistics
*Combinations and permutations
*History of mathematical notation

References

*

External links

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

{{Mathematical symbols notation language

Notation
Mathematical notation