Notation in probability and statistics

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Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

Probability theory[edit]

  • Random variables are usually written in upper case Roman letters: , , etc.
  • Particular realizations of a random variable are written in corresponding lower case letters. For example, could be a sample corresponding to the random variable . A cumulative probability is formally written to differentiate the random variable from its realization.[1]
  • The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and is short for , where is the event space and is a random variable. notation is used alternatively.
  • or indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as , while joint probability mass function or probability density function as and joint cumulative distribution function as .
  • or 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 uppercase calligraphic (e.g. for the set of sets on which we define the probability P)
  • Probability density functions (pdfs) and probability mass functions are denoted by lowercase letters, e.g. , or .
  • Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. , or .
  • Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:, or denoted as ,
  • In particular, the pdf of the standard normal distribution is denoted by , and its cdf by .
  • Some common operators:
  •  : expected value of X
  •  : variance of X
  •  : covariance of X and Y
  • X is independent of Y is often written or , and X is independent of Y given W is often written
  • , the conditional probability, is the probability of given [2]


  • Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).[3]
  • A tilde (~) denotes "has the probability distribution of".
  • Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an estimator of it, e.g., is an estimator for .
  • The arithmetic mean of a series of values is often denoted by placing an "overbar" over the symbol, e.g. , pronounced " bar".
  • Some commonly used symbols for sample statistics are given below:
  • Some commonly used symbols for population parameters are given below:
    • the population mean ,
    • the population variance ,
    • the population standard deviation ,
    • the population correlation ,
    • the population cumulants ,
  • is used for the order statistic, where is the sample minimum and is the sample maximum from a total sample size .[4]

Critical values[edit]

The α-level upper critical value of a probability distribution is the value exceeded with probability , that is, the value such that , where is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

  • or for the standard normal distribution
  • or for the t-distribution with degrees of freedom
  • or for the chi-squared distribution with degrees of freedom
  • or for the F-distribution with and degrees of freedom

Linear algebra[edit]

  • Matrices are usually denoted by boldface capital letters, e.g. .
  • Column vectors are usually denoted by boldface lowercase letters, e.g. .
  • The transpose operator is denoted by either a superscript T (e.g. ) or a prime symbol (e.g. ).
  • A row vector is written as the transpose of a column vector, e.g. or .


Common abbreviations include:

See also[edit]


  1. ^ "Calculating Probabilities from Cumulative Distribution Function". 2021-08-09. Retrieved 2024-02-26.
  2. ^ "Probability and stochastic processes", Applied Stochastic Processes, Chapman and Hall/CRC, pp. 9–36, 2013-07-22, ISBN 978-0-429-16812-3, retrieved 2023-12-08
  3. ^ "Letters of the Greek Alphabet and Some of Their Statistical Uses". 1999-02-13. Retrieved 2024-02-26.
  4. ^ "Order Statistics" (PDF). Retrieved 2024-02-26.
  • 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[edit]