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.

## 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 are usually written in upper case Roman letters, such as X or Y and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if P(X=x) is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), X, would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), x. It is important that X and x are not confused into meaning the same thing. X is an idea, x is a value. Clearly they are related, but they do not have identical meanings.
* Particular realisations 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. A cumulative probability is formally written P(X≤x) to distinguish the random variable from its realization.[1]
* The probability is sometimes written ℙ to distinguish it from other functions and measure P to avoid having to define "P is a probability" and ℙ(X∈A) is short for P({ω∈Ω:X(ω)∈A}), where Ω is the event space, X is a random variable that is a function of ω (i.e., it depends upon ω), and ω is some outcome of interest within the domain specified by Ω (say, a particular height, or a particular colour of a car). Pr⁡(A) notation is used alternatively.
* ℙ(A∩B) or ℙ[B∩A] indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as P(X,Y), while joint probability mass function or probability density function as f(x,y) and joint cumulative distribution function as F(x,y).
* ℙ(A∪B) or ℙ[B∪A] 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. ℱ 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. f(x), or fX(x).
* Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. F(x), or FX(x).
* Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:F‾(x)=1−F(x), or denoted as S(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

* X is independent of Y is often written X⊥Y or X⊥⊥Y, and X is independent of Y given W is often written

X⊥⊥Y|W or
X⊥Y|W

* P(A∣B), the conditional probability, is the probability of A given B [2]

## Statistics

* Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).[3]
* Some commonly used symbols for population or distribution parameters are given below:
* the population mean μ,
* the population variance σ2,
* the population standard deviation σ,
* the population correlation ρ,
* the population cumulants κr,
* 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., θ^ is an estimator for θ.
* The arithmetic mean of a series of values x1,x2,…,xn is often denoted by placing an "overbar" over the symbol, e.g. x¯, pronounced "x bar".
* Some commonly used symbols for sample statistics are given below:
* the sample mean x¯,
* the sample variance s2,
* the sample standard deviation s,
* the sample correlation coefficient r,
* the sample cumulants kr.
* x(k) is used for the kth order statistic, where x(1) is the sample minimum and x(n) is the sample maximum from a total sample size n.[4]

## 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:

* zα or z(α) for the standard normal distribution
* tα,ν or t(α,ν) for the t-distribution with ν degrees of freedom
* χα,ν2 or χ2(α,ν) for the chi-squared distribution with ν degrees of freedom
* Fα,ν1,ν2 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 lowercase 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:

* 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
* i.o. infinitely often, i.e. {An i.o.}=⋂N⋃n≥NAn
* ult. ultimately, i.e. {An ult.}=⋃N⋂n≥NAn

## See also

* Glossary of probability and statistics
* Combinations and permutations
* History of mathematical notation

## References

1. ↑ "Calculating Probabilities from Cumulative Distribution Function". 2021-08-09. https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/.
2. ↑ "Probability and stochastic processes", Applied Stochastic Processes (Chapman and Hall/CRC): pp. 9–36, 2013-07-22, doi:10.1201/b15257-3, ISBN 978-0-429-16812-3, http://dx.doi.org/10.1201/b15257-3, retrieved 2023-12-08
3. ↑ "Letters of the Greek Alphabet and Some of Their Statistical Uses". 1999-02-13. https://lesn.appstate.edu/olson/EDL7150/Components/Other useful links/Greek Alphabet and Statistics.htm.
4. ↑ "Order Statistics". https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf.

* 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

## External links

* Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.

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