In probability theory, the expected value of a random variable
X
{\displaystyle X}
, denoted
E
(
X
)
{\displaystyle \operatorname {E} (X)}
or
E
[
X
]
{\displaystyle \operatorname {E} [X]}
, is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of
X
{\displaystyle X}
. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.
By definition, the expected value of a constant random variable
X
=
c
{\displaystyle X=c}
is
c
{\displaystyle c}
. The expected value of a random variable
X
{\displaystyle X}
with equiprobable outcomes
{
c
1
,
…
,
c
n
}
{\displaystyle \{c_{1},\ldots ,c_{n}\}}
is defined as the arithmetic mean of the terms
c
i
.
{\displaystyle c_{i}.}
If some of the probabilities
Pr
(
X
=
c
i
)
{\displaystyle \Pr \,(X=c_{i})}
of an individual outcome
c
i
{\displaystyle c_{i}}
are unequal, then the expected value is defined to be the probability-weighted average of the
c
i
{\displaystyle c_{i}}
s, that is, the sum of the
n
{\displaystyle n}
products
c
i
⋅
Pr
(
X
=
c
i
)
{\displaystyle c_{i}\cdot \Pr \,(X=c_{i})}
. The expected value of a general random variable involves integration in the sense of Lebesgue.