Notation in probability and statistics

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

Probability theory

• Random variables are usually written in upper case Roman letters, such as ${\textstyle X}$ or ${\textstyle Y}$ 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 ${\displaystyle P(X=x)}$ 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), ${\textstyle x}$. It is important that ${\textstyle X}$ and ${\textstyle x}$ are not confused into meaning the same thing. ${\textstyle X}$ is an idea, ${\textstyle x}$ 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, ${\textstyle x_{1},x_{2},\ldots ,x_{n}}$ could be a sample corresponding to the random variable ${\textstyle X}$. A cumulative probability is formally written ${\displaystyle P(X\leq x)}$ to distinguish the random variable from its realization. ^[1]
• The probability is sometimes written ${\displaystyle \mathbb {P} }$ to distinguish it from other functions and measure P to avoid having to define "P is a probability" and ${\displaystyle \mathbb {P} (X\in A)}$ is short for ${\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}$, where ${\displaystyle \Omega }$ is the event space, ${\displaystyle X}$ is a random variable that is a function of ${\displaystyle \omega }$ (i.e., it depends upon ${\displaystyle \omega }$), and ${\displaystyle \omega }$ is some outcome of interest within the domain specified by ${\displaystyle \Omega }$ (say, a particular height, or a particular colour of a car). ${\displaystyle \Pr(A)}$ notation is used alternatively.
• ${\displaystyle \mathbb {P} (A\cap B)}$ or ${\displaystyle \mathbb {P} [B\cap A]}$ indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as ${\displaystyle P(X,Y)}$, while joint probability mass function or probability density function as ${\displaystyle f(x,y)}$ and joint cumulative distribution function as ${\displaystyle F(x,y)}$.
• ${\displaystyle \mathbb {P} (A\cup B)}$ or ${\displaystyle \mathbb {P} [B\cup A]}$ 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. ${\displaystyle {\mathcal {F}}}$ 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. ${\displaystyle f(x)}$, or ${\displaystyle f_{X}(x)}$.
• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. ${\displaystyle F(x)}$, or ${\displaystyle F_{X}(x)}$.
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:${\displaystyle {\overline {F}}(x)=1-F(x)}$, or denoted as ${\displaystyle S(x)}$,
• In particular, the pdf of the standard normal distribution is denoted by ${\textstyle \varphi (z)}$, and its cdf by ${\textstyle \Phi (z)}$.
• Some common operators:

• ${\textstyle \mathrm {E} [X]}$ : expected value of X
• ${\textstyle \operatorname {var} [X]}$ : variance of X
• ${\textstyle \operatorname {cov} [X,Y]}$ : covariance of X and Y

• X is independent of Y is often written ${\displaystyle X\perp Y}$ or ${\displaystyle X\perp \!\!\!\perp Y}$, and X is independent of Y given W is often written

${\displaystyle X\perp \!\!\!\perp Y\,|\,W}$ or

${\displaystyle X\perp Y\,|\,W}$

• ${\displaystyle \textstyle P(A\mid B)}$, the conditional probability , is the probability of ${\displaystyle \textstyle A}$given${\displaystyle \textstyle B}$ ^[2]

Statistics

• 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., ${\displaystyle {\widehat {\theta }}}$ is an estimator for ${\displaystyle \theta }$.
• The arithmetic mean of a series of values ${\textstyle x_{1},x_{2},\ldots ,x_{n}}$ is often denoted by placing an "overbar" over the symbol, e.g. ${\displaystyle {\bar {x}}}$, pronounced "${\textstyle x}$ bar".
• Some commonly used symbols for sample statistics are given below:
  • the sample mean ${\displaystyle {\bar {x}}}$,
  • the sample variance ${\textstyle s^{2}}$,
  • the sample standard deviation ${\textstyle s}$,
  • the sample correlation coefficient ${\textstyle r}$,
  • the sample cumulants ${\textstyle k_{r}}$.
• Some commonly used symbols for population parameters are given below:
  • the population mean ${\textstyle \mu }$,
  • the population variance ${\textstyle \sigma ^{2}}$,
  • the population standard deviation ${\textstyle \sigma }$,
  • the population correlation ${\textstyle \rho }$,
  • the population cumulants ${\textstyle \kappa _{r}}$,
• ${\displaystyle x_{(k)}}$ is used for the ${\displaystyle k^{\text{th}}}$ order statistic, where ${\displaystyle x_{(1)}}$ is the sample minimum and ${\displaystyle x_{(n)}}$ is the sample maximum from a total sample size ${\textstyle n}$. ^[4]

Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability ${\textstyle \alpha }$, that is, the value ${\textstyle x_{\alpha }}$ such that ${\textstyle F(x_{\alpha })=1-\alpha }$, where ${\textstyle F}$ is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

• ${\textstyle z_{\alpha }}$ or ${\textstyle z(\alpha )}$ for the standard normal distribution
• ${\textstyle t_{\alpha ,\nu }}$ or ${\textstyle t(\alpha ,\nu )}$ for the t-distribution with ${\textstyle \nu }$ degrees of freedom
• ${\displaystyle {\chi _{\alpha ,\nu }}^{2}}$ or ${\displaystyle {\chi }^{2}(\alpha ,\nu )}$ for the chi-squared distribution with ${\textstyle \nu }$ degrees of freedom
• ${\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}$ or ${\textstyle F(\alpha ,\nu _{1},\nu _{2})}$ for the F-distribution with ${\textstyle \nu _{1}}$ and ${\textstyle \nu _{2}}$ degrees of freedom

Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. ${\textstyle {\mathbf {A}}}$.
• Column vectors are usually denoted by boldface lowercase letters, e.g. ${\textstyle {\mathbf {x}}}$.
• The transpose operator is denoted by either a superscript T (e.g. ${\textstyle {\mathbf {A}}^{\mathrm {T} }}$) or a prime symbol (e.g. ${\textstyle {\mathbf {A}}'}$).
• A row vector is written as the transpose of a column vector, e.g. ${\textstyle {\mathbf {x}}^{\mathrm {T} }}$ or ${\textstyle {\mathbf {x}}'}$.

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 ${\displaystyle \nu }$)
• 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. ${\displaystyle \{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}}$
• ult. ultimately, i.e. ${\displaystyle \{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}}$

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. Retrieved 2024-02-26.
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 , retrieved 2023-12-08
3. "Letters of the Greek Alphabet and Some of Their Statistical Uses". les.appstate.edu/. 1999-02-13. Retrieved 2024-02-26.
4. "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

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