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: , , etc.
 Particular realizations of a random variable are written in corresponding lower case letters. For example,  could be a sample corresponding to the random variable                                                                                                                                                                                            . A cumulative probability is formally written  to differentiate the random variable from its realization.
 The probability is sometimes written  to distinguish it from other functions and measure P so as to avoid having to define "P is a probability" and  is short for , where  is the event space and  is a random variable.  notation is used alternatively.
 or  indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as , while joint probability mass function or probability density function as  and joint cumulative distribution function as .
 or  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.  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. , or .
Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. , or .
 Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:, or denoted as ,
In particular, the pdf of the standard normal distribution is denoted by , and its cdf by .
Some common operators: 
  : expected value of X
  : variance of X
  : covariance of X and Y
 X is independent of Y is often written  or , and X is independent of Y given W is often written 
 or

 , the conditional probability, is the probability of  given , i.e.,  after  is observed.

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.,  is an estimator for .
The arithmetic mean of a series of values  is often denoted by placing an "overbar" over the symbol, e.g. , pronounced " bar".
Some commonly used symbols for sample statistics are given below:
the sample mean ,
the sample variance ,
 the sample standard deviation ,
the sample correlation coefficient ,
the sample cumulants .
Some commonly used symbols for population parameters are given below:
the population mean ,
the population variance ,
 the population standard deviation ,
the population correlation ,
the population cumulants ,
 is used for the  order statistic, where  is the sample minimum and  is the sample maximum from a total sample size .

Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability , that is, the value  such that , where  is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
 or  for the standard normal distribution
 or  for the t-distribution with  degrees of freedom
 or  for the chi-squared distribution with  degrees of freedom
 or  for the F-distribution with  and  degrees of freedom

Linear algebra

Matrices are usually denoted by boldface capital letters, e.g. . 
Column vectors are usually denoted by boldface lowercase letters, e.g. .
The transpose operator is denoted by either a superscript T (e.g. ) or a prime symbol (e.g. ).
A row vector is written as the transpose of a column vector, e.g.  or .

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 )
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. 
 ult. ultimately, i.e.

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.

 Notation
Mathematical notation