Notation in probability and statistics
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.
- 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
- 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 , i.e., after is observed.
- 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 .
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:
- 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
- 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:
- 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.
- Glossary of probability and statistics
- Combinations and permutations
- History of mathematical notation
- 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
- Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.