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, …, x_n could be a sample corresponding to the random variable X and a cumulative probability is formally written $P(X>x)$ to differentiate random variable from 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 $\mathbb{P}(A)$ is short for $P(\{\omega: X(\omega) \in A\})$, where $\omega$ is an event and $X(\omega)$ a corresponding random variable.
• $\mathbb{P}(A \cap B)$ or $\mathbb{P}[A \cap B]$ indicates the probability that events A and B both occur.
• $\mathbb{P}(A \cup B)$ or $\mathbb{P}[A \cup B]$ 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. $\mathcal F$ 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:$\overline{F}(x) =1-F(x)$
• In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
• Some common operators:

• 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 posterior 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., $\widehat{\theta}$ is an estimator for $\theta$.
• The arithmetic mean of a series of values x₁, x₂, ..., x_n 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:
  • the sample mean $\bar{x}$,
  • the sample variance s²,
  • 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 σ²,
  • the population standard deviation σ,
  • the population correlation ρ,
  • the population cumulants κ_r.

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
• ${\chi_{\alpha,\nu}}^{2}$ 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 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

See also

• Glossary of probability and statistics
• Combinations and permutations
• Typographical conventions in mathematical formulae
• History of mathematical notation

External links

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