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 $P(X=x)$ 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 $P(X\leq x)$ to distinguish the random variable from its realization.
• The probability is sometimes written $\mathbb {P}$ to distinguish it from other functions and measure P to avoid having to define "P is a probability" and $\mathbb {P} (X\in A)$ is short for $P(\{\omega \in \Omega :X(\omega )\in A\})$, where $\Omega$ is the event space, $X$ is a random variable that is a function of $\omega$ (i.e., it depends upon $\omega$), and $\omega$ is some outcome of interest within the domain specified by $\Omega$ (say, a particular height, or a particular colour of a car). $\Pr(A)$ notation is used alternatively.
• $\mathbb {P} (A\cap B)$ or $\mathbb {P} [B\cap A]$ indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as $P(X,Y)$, while joint probability mass function or probability density function as $f(x,y)$ and joint cumulative distribution function as $F(x,y)$.
• $\mathbb {P} (A\cup B)$ or $\mathbb {P} [B\cup A]$ 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. ${\mathcal {F}}$ 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. $f(x)$, or $f_{X}(x)$.
• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. $F(x)$, or $F_{X}(x)$.
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:${\overline {F}}(x)=1-F(x)$, or denoted as $S(x)$,
• 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 $X\perp Y$ or $X\perp \!\!\!\perp Y$, and X is independent of Y given W is often written

$X\perp \!\!\!\perp Y\,|\,W$ or

$X\perp Y\,|\,W$

• $\textstyle P(A\mid B)$, the conditional probability, is the probability of $\textstyle A$ given $\textstyle B$

Statistics

• 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 (also known as a circumflex), over a true parameter denotes an estimator of it, e.g., ${\widehat {\theta }}$ is an estimator for $\theta$.
• 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. ${\bar {x}}$, pronounced "${\textstyle x}$ bar".
• Some commonly used symbols for sample statistics are given below:
  • the sample mean ${\bar {x}}$,
  • 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 ${\textstyle \kappa _{r}}$,
• $x_{(k)}$ is used for the $k^{\text{th}}$ order statistic, where $x_{(1)}$ is the sample minimum and $x_{(n)}$ is the sample maximum from a total sample size ${\textstyle n}$.

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
• ${\chi _{\alpha ,\nu }}^{2}$ or ${\chi }^{2}(\alpha ,\nu )$ for the chi-squared distribution with ${\textstyle \nu }$ degrees of freedom
• $F_{\alpha ,\nu _{1},\nu _{2}}$ 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 $\nu$)
• 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. $\{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}$
• ult. ultimately, i.e. $\{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}$

See also

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

References

• 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.