# Notation in probability and statistics

Probability |
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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.
- The probability is sometimes written to distinguish it from other functions and measure
*P*so as 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*

- : expected value of

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

- or

- , the
*conditional probability*, is the probability of*given*, i.e.,*after*is observed.^{[citation needed]}

## Statistics[edit]

- Greek letters (e.g.
*θ*,*β*) are commonly used to denote unknown parameters (population parameters). - A tilde (~) denotes "has the probability distribution of".
- Placing a hat, or caret, 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:
- the sample mean ,
- the sample variance ,
- the sample standard deviation
*,* - the sample correlation coefficient
*,* - the sample cumulants .

- 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 .

## Critical values[edit]

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:

- 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****.**

## Abbreviations[edit]

Common abbreviations include:

**a.e.**almost everywhere**a.s.**almost surely**cdf**cumulative distribution function**cmf**cumulative mass function**df**degrees of freedom (also )**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.**ult.**ultimately, i.e.

## See also[edit]

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

## References[edit]

- 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]

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