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: $X$, $Y$ , etc.
 * Particular realizations 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 differentiate 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 and $$X(\omega)$$ is a random variable. $$\Pr(A)$$ 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 $$\textstyle B$$

Statistics

 * Greek letters (e.g. &theta;, &beta;) 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 $x_1,x_2, \ldots,x_n$ 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 $n$.

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