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.

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 χ²(α,ν) for the chi-square 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 ν)
• 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

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.