Notation in probability and statistics

From Wikipedia, the free encyclopedia

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

[edit] Probability theory

Random variables (e.g. the height of students) are written in upper case.
Singular values are written in lower case (e.g. P( X = x ) can be the probability that a student is of height x).
$P(A \cap B)$ indicates the probability that events A and B both occur.
$P(A \cup B)$ 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)
$N \choose k$ (N-choose-k) is defined as the number of ways in which one can select k objects from N objects, and is an alternative term for binomial coefficient. Also defined as combination without repetition in combinations and permutations.
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

[edit] 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".
The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is denoted by $\bar{x}$ , pronounced "x bar".

[edit] 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 $χ 2 (α,ν)$ 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

[edit] 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′.

Note: Wikipedia articles usually use the superscript T to denote transpose. The prime symbol is more difficult to produce and is rather small in the default font.