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.

Contents 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 (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).

·      indicates the probability that events A and B both occur.

·      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. for the set of sets on which we define the probability P)

·      (N-choose-k) is defined as a 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

 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. , pronounced "theta hat".

·      The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is denoted by , 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

·      or χ²(α,ν) for the chi-square distribution with ν degrees of freedom

·      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′.

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.

 Abbreviations

Common abbreviations include:

·      a.e. almost everywhere

·      cdf cumulative distribution function

·      df degrees of freedom

·      pdf probability density function

·      pmf probability mass function

·      r.v. random variable

 See also

·      Glossary of probability and statistics

·      Combinations and permutations

·      Typographical conventions in