Notation in probability

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.
* The probability is sometimes written $\mathbb$ to distinguish it from other functions and measure ''P'' to avoid having to define "''P'' is a probability" and $\mathbb(X\in A)$ is short for $P(\)$, 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.
*$\mathbb(A \cap B)$ or \mathbb \cap A/math> 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 $F(x, y)$.
*$\mathbb(A \cup B)$ or \mathbb \cup A/math> 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:$\overline(x) =1-F(x)$, 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 /math>: expected value of ''X''
:* \operatorname /math>: variance of ''X''
:* \operatorname ,Y/math>: 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., $\widehat$ 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. $\bar$, pronounced "$x$ bar".
*Some commonly used symbols for sample statistics are given below:
**the sample mean $\bar$,
**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 ''$\kappa_r$'',
*$x_$ is used for the $k^\text$ order statistic, where $x_$ is the sample minimum and $x_$ is the sample maximum from a total sample size $n$.

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
*$t_$ or $t(\alpha,\nu)$ for the ''t''-distribution with $\nu$ degrees of freedom
*$^2$ or $^(\alpha,\nu)$ for the chi-squared distribution with $\nu$ degrees of freedom
*$F_$ 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. $\bold$.
*Column vectors are usually denoted by boldface lowercase letters, e.g. $\bold$.
*The transpose operator is denoted by either a superscript T (e.g. $\bold^\mathrm$) or a prime symbol (e.g. $\bold'$).
*A row vector is written as the transpose of a column vector, e.g. $\bold^\mathrm$ or $\bold'$.

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. $\ = \bigcap_N\bigcup_ A_n$
* ult. ultimately, i.e. $\ = \bigcup_N\bigcap_ A_n$

See also

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

References

*

External links

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

{{Mathematical symbols notation language

Notation
Mathematical notation