Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.
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 x1, x2, …, xn could be a sample corresponding to the random variable X and a cumulative probability is formally written
P
(
X
>
x
)
to differentiate random variable from realization.
The probability is sometimes written
P
to distinguish it from other functions and measure P so as to avoid having to define " P is a probability" and
P
(
A
)
is short for
P
(
{
ω
:
X
(
ω
)
∈
A
}
)
, where
ω
is an event and
X
(
ω
)
a corresponding random variable.
P
(
A
∩
B
)
or
P
[
A
∩
B
]
indicates the probability that events A and B both occur.
P
(
A
∪
B
)
or
P
[
A
∪
B
]
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 upper case calligraphic (e.g.
F
for the set of sets on which we define the probability P)
Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:
F
¯
(
x
)
=
1
−
F
(
x
)
In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
Some common operators:
E[X] : expected value of X
var[X] : variance of X
cov[X, Y] : covariance of X and Y
X is independent of Y is often written
X
⊥
Y
or
X
⊥
⊥
Y
, and X is independent of Y given W is often written
X
⊥
⊥
Y
|
W
or
X
⊥
Y
|
W
P
(
A
∣
B
)
, the posterior probability, is the probability of
A
given
B
, i.e.,
A
after
B
is observed.
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, over a true parameter denotes an estimator of it, e.g.,
θ
^
is an estimator for
θ
.
The arithmetic mean of a series of values x1, x2, ..., xn is often denoted by placing an "overbar" over the symbol, e.g.
x
¯
, pronounced "x bar".
Some commonly used symbols for sample statistics are given below:
the sample mean
x
¯
,
the sample variance s2,
the sample standard deviation s,
the sample correlation coefficient r,
the sample cumulants kr.
Some commonly used symbols for population parameters are given below:
the population mean μ,
the population variance σ2,
the population standard deviation σ,
the population correlation ρ,
the population cumulants κr.
The α-level upper critical value of a probability distribution is the value exceeded with probability α, that is, the value xα such that F(xα) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
zα or z(α) for the Standard normal distribution
tα,ν or t(α,ν) for the t-distribution with ν degrees of freedom
χ
α
,
ν
2
or
χ
2
(
α
,
ν
)
for the chi-squared distribution with ν degrees of freedom
F
α
,
ν
1
,
ν
2
or F(α,ν1,ν2) for the F-distribution with ν1 and ν2 degrees of freedom
Matrices are usually denoted by boldface capital letters, e.g. A.
Column vectors are usually denoted by boldface lower case letters, e.g. x.
The transpose operator is denoted by either a superscript T (e.g. AT) or a prime symbol (e.g. A′).
A row vector is written as the transpose of a column vector, e.g. xT or x′.
Common abbreviations include:
a.e. almost everywhere
a.s. almost surely
cdf cumulative distribution function
cmf cumulative mass function
df degrees of freedom (also
ν
)
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
