Notation in probability and statistics

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

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, …, xn 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:

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 kr.
  • 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 κr.
  • The arithmetic mean of a set of numbers x1, x2, ..., xn 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 xα such that F(xα) = 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:

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. AT) or a prime symbol (e.g. A′).
  • A row vector is written as the transpose of a column vector, e.g. xT or x′.

Abbreviations

Common abbreviations include:

See also

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 

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