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, such as or 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 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), . It is important that and are not confused into meaning the same thing. is an idea, 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, could be a sample corresponding to the random variable . A cumulative probability is formally written to distinguish 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, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular height, or a particular colour of a car). 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
- or
- , the conditional probability, is the probability of given [2]
Statistics
[edit]- 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:
- 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 .[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 .
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]- ^ "Calculating Probabilities from Cumulative Distribution Function". 2021-08-09. Retrieved 2024-02-26.
- ^ "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
- ^ "Letters of the Greek Alphabet and Some of Their Statistical Uses". les.appstate.edu/. 1999-02-13. Retrieved 2024-02-26.
- ^ "Order Statistics" (PDF). colorado.edu. 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]- Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.