Notation in probability and statistics

Probability
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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: ${\displaystyle X}$, ${\displaystyle Y}$, etc.
• Particular realizations of a random variable are written in corresponding lower case letters. For example, ${\displaystyle x_1,x_2,\dots ,x_n}$ could be a sample corresponding to the random variable ${\displaystyle X}$. A cumulative probability is formally written ${\displaystyle P(X\le x)}$ to differentiate the random variable from its realization.
• The probability is sometimes written ${\displaystyle \mathbb{P}}$ to distinguish it from other functions and measure P so as to avoid having to define "P is a probability" and ${\displaystyle \mathbb{P}(X\in A)}$ is short for ${\displaystyle P(\{\omega \in \mathrm{\Omega }:X(\omega )\in A\})}$, where ${\displaystyle \mathrm{\Omega }}$ is the event space and ${\displaystyle X(\omega )}$ is a random variable. ${\displaystyle Pr(A)}$ 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:${\displaystyle \bar{F}(x)=1-F(x)}$, or denoted as ${\displaystyle S(x)}$,
• In particular, the pdf of the standard normal distribution is denoted by ${\displaystyle \phi (z)}$, and its cdf by ${\displaystyle \mathrm{\Phi }(z)}$.
• Some common operators:

• ${\displaystyle \mathrm{E}[X]}$ : expected value of X
• ${\displaystyle \mathrm{var}[X]}$ : variance of X
• ${\displaystyle \mathrm{cov}[X,Y]}$ : 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

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

• ${\displaystyle P(A\mid B)}$, the conditional probability, is the probability of ${\displaystyle A}$ given ${\displaystyle B}$ ^[1]

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., ${\displaystyle \hat{\theta }}$ is an estimator for ${\displaystyle \theta }$.
• The arithmetic mean of a series of values ${\displaystyle x_1,x_2,\dots ,x_n}$ is often denoted by placing an "overbar" over the symbol, e.g. ${\displaystyle \overline{x}}$, pronounced "${\displaystyle x}$ bar".
• Some commonly used symbols for sample statistics are given below:
  • the sample mean ${\displaystyle \overline{x}}$,
  • the sample variance ${\displaystyle s^2}$,
  • the sample standard deviation ${\displaystyle s}$,
  • the sample correlation coefficient ${\displaystyle r}$,
  • the sample cumulants ${\displaystyle k_r}$.
• Some commonly used symbols for population parameters are given below:
  • the population mean ${\displaystyle \mu }$,
  • the population variance ${\displaystyle {\sigma }^2}$,
  • the population standard deviation ${\displaystyle \sigma }$,
  • the population correlation ${\displaystyle \rho }$,
  • the population cumulants ${\displaystyle {\kappa }_r}$,
• ${\displaystyle x_{(k)}}$ 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 ${\displaystyle n}$.

Critical values

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

• ${\displaystyle z_{\alpha }}$ or ${\displaystyle z(\alpha )}$ for the standard normal distribution
• ${\displaystyle t_{\alpha ,\nu }}$ or ${\displaystyle t(\alpha ,\nu )}$ for the t-distribution with ${\displaystyle \nu }$ degrees of freedom
• ${\displaystyle {{\chi }_{\alpha ,\nu }}^2}$ or ${\displaystyle {\chi }^2(\alpha ,\nu )}$ for the chi-squared distribution with ${\displaystyle \nu }$ degrees of freedom
• ${\displaystyle F_{\alpha ,{\nu }_1,{\nu }_2}}$ or ${\displaystyle F(\alpha ,{\nu }_1,{\nu }_2)}$ for the F-distribution with ${\displaystyle {\nu }_1}$ and ${\displaystyle {\nu }_2}$ degrees of freedom

Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. ${\displaystyle \text{bold}A}$.
• Column vectors are usually denoted by boldface lowercase letters, e.g. ${\displaystyle \text{bold}x}$.
• The transpose operator is denoted by either a superscript T (e.g. ${\displaystyle \text{bold}{A}^{\mathrm{T}}}$) or a prime symbol (e.g. ${\displaystyle \text{bold}{A}^{\prime }}$).
• A row vector is written as the transpose of a column vector, e.g. ${\displaystyle \text{bold}{x}^{\mathrm{T}}}$ or ${\displaystyle \text{bold}{x}^{\prime }}$.

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. ${\displaystyle \{A_n\text{ i.o.}\}=\bigcap _N\bigcup _{n\ge N}{A}_n}$
• ult. ultimately, i.e. ${\displaystyle \{A_n\text{ ult.}\}=\bigcup _N\bigcap _{n\ge N}{A}_n}$

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.

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