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 $X$ or $Y$ 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), $x$ . It is important that $X$ and $x$ are not confused into meaning the same thing. $X$ is an idea, $x$ 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, $x_1,x_2, \ldots,x_n$ could be a sample corresponding to the random variable $X$ . A cumulative probability is formally written $P(X\le x)$ to distinguish the random variable from its realization.{{Cite web |date=2021-08-09 |title=Calculating Probabilities from Cumulative Distribution Function |url=https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/ |access-date=2024-02-26}}
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 $\varphi(z)$ , and its cdf by $\Phi(z)$ .
Some common operators:

:* $\mathrm{E}[X]$ : expected value of X

:* $\operatorname{var}[X]$ : variance of X

:* $\operatorname{cov}[X,Y]$ : 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 $\textstyle B$ {{Citation |title=Probability and stochastic processes |date=2013-07-22 |url=http://dx.doi.org/10.1201/b15257-3 |work=Applied Stochastic Processes |pages=9–36 |access-date=2023-12-08 |publisher=Chapman and Hall/CRC |doi=10.1201/b15257-3 |isbn=978-0-429-16812-3}}

Statistics

Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).{{Cite web |date=1999-02-13 |title=Letters of the Greek Alphabet and Some of Their Statistical Uses |url=https://lesn.appstate.edu/olson/EDL7150/Components/Other%20useful%20links/Greek%20Alphabet%20and%20Statistics.htm |access-date=2024-02-26 |website=les.appstate.edu/}}
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 $x_1,x_2, \ldots,x_n$ is often denoted by placing an "overbar" over the symbol, e.g. $\bar{x}$ , pronounced " $x$ bar".
Some commonly used symbols for sample statistics are given below:
the sample mean $\bar{x}$ ,
the sample variance $s^2$ ,
the sample standard deviation $s$ ,
the sample correlation coefficient $r$ ,
the sample cumulants $k_r$ .
Some commonly used symbols for population parameters are given below:
the population mean $\mu$ ,
the population variance $\sigma^2$ ,
the population standard deviation $\sigma$ ,
the population correlation $\rho$ ,
the population cumulants $\kappa_r$ ,
$x_{(k)}$ 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 $n$ .{{Cite web |title=Order Statistics |url=https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf |access-date=2024-02-26 |website=colorado.edu}}

Critical values

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

$z_\alpha$ or $z(\alpha)$ for the standard normal distribution
$t_{\alpha,\nu}$ or $t(\alpha,\nu)$ for the t-distribution with $\nu$ degrees of freedom
${\chi_{\alpha,\nu}}^2$ or ${\chi}^{2}(\alpha,\nu)$ for the chi-squared distribution with $\nu$ degrees of freedom
$F_{\alpha,\nu_1,\nu_2}$ or $F(\alpha,\nu_1,\nu_2)$ for the F-distribution with $\nu_1$ and $\nu_2$ degrees of freedom

Linear algebra

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

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. $\{ A_n \text{ ult.} \} = \bigcup_N\bigcap_{n\geq N} A_n$

References

{{Citation |title = Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation |first1 = Max |last1 = Halperin |first2 = H. O. |last2 = Hartley |first3 = P. G. |last3 = Hoel |journal = The American Statistician |volume = 19 |year = 1965 |pages = 12–14 |issue = 3 |doi = 10.2307/2681417 |jstor = 2681417 }}

External links

[http://jeff560.tripod.com/stat.html Earliest Uses of Symbols in Probability and Statistics], maintained by Jeff Miller.

Notation

Category:Mathematical notation