Notation in probability and statistics

Probability theory and statistics has some commonly used conventions of its own, in addition to standard mathematical notation and mathematical symbols.

1 Probability theory
2 Statistics
3 Critical values
4 Linear algebra
5 Abbreviations
6 See also
7 References
8 External links

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, …, x_n could be a sample corresponding to the random variable X.
$P(A \cap B)$ or $P[A \cap B]$ indicates the probability that events A and B both occur.
$P(A \cup B)$ or $P[A \cup B]$ 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. $\mathcal F$ 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).
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

Statistics

Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
An estimate of a parameter is often denoted by placing a caret over the corresponding symbol, e.g. $\hat{\theta}$ , pronounced "theta hat".
Some commonly used symbols for sample statistics are given below:
- the sample mean $\bar{x}$ ,
- the sample variance s²,
- 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 σ²,
- the population correlation ρ,
- the population cumulants κ_r.
The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is denoted by $\bar{x}$ , pronounced "x bar".

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
$\chi^2_{\alpha,\nu}$ 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 lower case 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

References

Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", The American Statistician 19 (3): 12–14, doi:10.2307/2681417, JSTOR 2681417

External links

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