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 <math display="inline">X</math> or <math display="inline">Y</math> 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 <math>P(X = x) </math> 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), <math display="inline">x</math>. It is important that <math display="inline">X</math> and <math display="inline">x</math> are not confused into meaning the same thing. <math display="inline">X</math> is an idea, <math display="inline">x</math> 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, <math display="inline">x_1,x_2, \ldots,x_n</math> could be a sample corresponding to the random variable <math display="inline">X</math>. A cumulative probability is formally written <math>P(X\le x) </math> to distinguish the random variable from its realization.
The probability is sometimes written <math>\mathbb{P} </math> to distinguish it from other functions and measure P to avoid having to define "P is a probability" and <math>\mathbb{P}(X\in A) </math> is short for <math>P(\{\omega \in\Omega: X(\omega) \in A\})</math>, where <math>\Omega</math> is the sample space, <math>X</math> is a random variable that is a function of <math>\omega</math> (i.e., it depends upon <math>\omega</math>), and <math>\omega</math> is some outcome of interest within the domain specified by <math>\Omega</math> (say, a particular height, or a particular colour of a car). <math>\Pr(A)</math> notation is used alternatively.
<math>\mathbb{P}(A \cap B)</math> or <math>\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>P(X, Y)</math>, while joint probability mass function or probability density function as <math>f(x, y)</math> and joint cumulative distribution function as <math>F(x, y)</math>.
<math>\mathbb{P}(A \cup B)</math> or <math>\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>\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>f(x)</math>, or <math>f_X(x)</math>.
Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. <math>F(x)</math>, or <math>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>\overline{F}(x) =1-F(x)</math>, or denoted as <math>S(x)</math>,
In particular, the pdf of the standard normal distribution is denoted by <math display="inline">\varphi(z)</math>, and its cdf by <math display="inline">\Phi(z)</math>.
Some common operators:

:* <math display="inline">\mathrm{E}[X]</math>: expected value of X

:* <math display="inline">\operatorname{var}[X]</math>: variance of X

:* <math display="inline">\operatorname{cov}[X,Y]</math>: covariance of X and Y

X is independent of Y is often written <math>X \perp Y</math> or <math>X \perp\!\!\!\perp Y</math>, and X is independent of Y given W is often written

:<math>X \perp\!\!\!\perp Y \,|\, W </math> or

:<math>X \perp Y \,|\, W</math>

<math>\textstyle P(A\mid B)</math>, the conditional probability, is the probability of <math>\textstyle A</math> given <math>\textstyle B</math>

Statistics

Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
Some commonly used symbols for population or distribution parameters are given below:
the population mean <math display="inline">\mu</math>,
the population variance <math display="inline">\sigma^2</math>,
the population standard deviation <math display="inline">\sigma</math>,
the population correlation <math display="inline">\rho</math>,
the population cumulants <math display="inline">\kappa_r</math>,
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>\widehat{\theta}</math> is an estimator for <math>\theta</math>.
The arithmetic mean of a series of values <math display="inline">x_1,x_2, \ldots,x_n</math> is often denoted by placing an "overbar" over the symbol, e.g. <math>\bar{x}</math>, pronounced "<math display="inline">x</math> bar".
Some commonly used symbols for sample statistics are given below:
the sample mean <math>\bar{x}</math>,
the sample variance <math display="inline">s^2</math>,
the sample standard deviation <math display="inline">s</math>,
the sample correlation coefficient <math display="inline">r</math>,
the sample cumulants <math display="inline">k_r</math>.
<math>x_{(k)}</math> is used for the <math>k^\text{th}</math> order statistic, where <math>x_{(1)}</math> is the sample minimum and <math>x_{(n)}</math> is the sample maximum from a total sample size <math display="inline">n</math>.

Critical values

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

<math display="inline">z_\alpha</math> or <math display="inline">z(\alpha)</math> for the standard normal distribution
<math display="inline">t_{\alpha,\nu}</math> or <math display="inline">t(\alpha,\nu)</math> for the t-distribution with <math display="inline">\nu</math> degrees of freedom
<math>{\chi_{\alpha,\nu^2</math> or <math>{\chi}^{2}(\alpha,\nu)</math> for the chi-squared distribution with <math display="inline">\nu</math> degrees of freedom
<math>F_{\alpha,\nu_1,\nu_2}</math> or <math display="inline">F(\alpha,\nu_1,\nu_2)</math> for the F-distribution with <math display="inline">\nu_1</math> and <math display="inline">\nu_2</math> degrees of freedom

Linear algebra

Matrices are usually denoted by boldface capital letters, e.g. <math display="inline">\bold{A}</math>.
Column vectors are usually denoted by boldface lowercase letters, e.g. <math display="inline">\bold{x}</math>.
The transpose operator is denoted by either a superscript T (e.g. <math display="inline">\bold{A}^\mathrm{T}</math>) or a prime symbol (e.g. <math display="inline">\bold{A}'</math>).
A row vector is written as the transpose of a column vector, e.g. <math display="inline">\bold{x}^\mathrm{T}</math> or <math display="inline">\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>\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> \{ A_n\text{ i.o.} \} = \bigcap_N\bigcup_{n\geq N} A_n </math>
ult. ultimately, i.e. <math>\{ A_n \text{ ult.} \} = \bigcup_N\bigcap_{n\geq N} A_n </math>

References

External links

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