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: [math]\displaystyle{ X }[/math], [math]\displaystyle{ Y }[/math], etc.
• Particular realizations of a random variable are written in corresponding lower case letters. For example, [math]\displaystyle{ x_1,x_2, \ldots,x_n }[/math] could be a sample corresponding to the random variable [math]\displaystyle{ X }[/math]. A cumulative probability is formally written [math]\displaystyle{ P(X\le x) }[/math] to differentiate the random variable from its realization.
• The probability is sometimes written [math]\displaystyle{ \mathbb{P} }[/math] to distinguish it from other functions and measure P so as to avoid having to define "P is a probability" and [math]\displaystyle{ \mathbb{P}(X\in A) }[/math] is short for [math]\displaystyle{ P(\{\omega \in\Omega: X(\omega) \in A\}) }[/math], where [math]\displaystyle{ \Omega }[/math] is the event space and [math]\displaystyle{ X(\omega) }[/math] is a random variable. [math]\displaystyle{ \Pr(A) }[/math] notation is used alternatively.
• [math]\displaystyle{ \mathbb{P}(A \cap B) }[/math] or [math]\displaystyle{ \mathbb{P}[B \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 [math]\displaystyle{ P(X, Y) }[/math], while joint probability mass function or probability density function as [math]\displaystyle{ f(x, y) }[/math] and joint cumulative distribution function as [math]\displaystyle{ F(x, y) }[/math].
• [math]\displaystyle{ \mathbb{P}(A \cup B) }[/math] or [math]\displaystyle{ \mathbb{P}[B \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. [math]\displaystyle{ \mathcal F }[/math] 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. [math]\displaystyle{ f(x) }[/math], or [math]\displaystyle{ f_X(x) }[/math].
• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. [math]\displaystyle{ F(x) }[/math], or [math]\displaystyle{ F_X(x) }[/math].
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:[math]\displaystyle{ \overline{F}(x) =1-F(x) }[/math], or denoted as [math]\displaystyle{ S(x) }[/math],
• In particular, the pdf of the standard normal distribution is denoted by [math]\displaystyle{ \varphi(z) }[/math], and its cdf by [math]\displaystyle{ \Phi(z) }[/math].
• Some common operators:

• [math]\displaystyle{ \mathrm{E}[X] }[/math] : expected value of X
• [math]\displaystyle{ \operatorname{var}[X] }[/math] : variance of X
• [math]\displaystyle{ \operatorname{cov}[X,Y] }[/math] : covariance of X and Y

• X is independent of Y is often written [math]\displaystyle{ X \perp Y }[/math] or [math]\displaystyle{ X \perp\!\!\!\perp Y }[/math], and X is independent of Y given W is often written

[math]\displaystyle{ X \perp\!\!\!\perp Y \,|\, W }[/math] or

[math]\displaystyle{ X \perp Y \,|\, W }[/math]

• [math]\displaystyle{ \textstyle P(A\mid B) }[/math], the conditional probability, is the probability of [math]\displaystyle{ \textstyle A }[/math] given [math]\displaystyle{ \textstyle B }[/math] ^[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., [math]\displaystyle{ \widehat{\theta} }[/math] is an estimator for [math]\displaystyle{ \theta }[/math].
• The arithmetic mean of a series of values [math]\displaystyle{ x_1,x_2, \ldots,x_n }[/math] is often denoted by placing an "overbar" over the symbol, e.g. [math]\displaystyle{ \bar{x} }[/math], pronounced "[math]\displaystyle{ x }[/math] bar".
• Some commonly used symbols for sample statistics are given below:
  • the sample mean [math]\displaystyle{ \bar{x} }[/math],
  • the sample variance [math]\displaystyle{ s^2 }[/math],
  • the sample standard deviation [math]\displaystyle{ s }[/math],
  • the sample correlation coefficient [math]\displaystyle{ r }[/math],
  • the sample cumulants [math]\displaystyle{ k_r }[/math].
• Some commonly used symbols for population parameters are given below:
  • the population mean [math]\displaystyle{ \mu }[/math],
  • the population variance [math]\displaystyle{ \sigma^2 }[/math],
  • the population standard deviation [math]\displaystyle{ \sigma }[/math],
  • the population correlation [math]\displaystyle{ \rho }[/math],
  • the population cumulants [math]\displaystyle{ \kappa_r }[/math],
• [math]\displaystyle{ x_{(k)} }[/math] is used for the [math]\displaystyle{ k^\text{th} }[/math] order statistic, where [math]\displaystyle{ x_{(1)} }[/math] is the sample minimum and [math]\displaystyle{ x_{(n)} }[/math] is the sample maximum from a total sample size [math]\displaystyle{ n }[/math].

Critical values

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

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

Linear algebra

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

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 [math]\displaystyle{ \nu }[/math])
• 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. [math]\displaystyle{ \{ A_n\text{ i.o.} \} = \bigcap_N\bigcup_{n\geq N} A_n }[/math]
• ult. ultimately, i.e. [math]\displaystyle{ \{ A_n \text{ ult.} \} = \bigcup_N\bigcap_{n\geq N} A_n }[/math]

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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