# What is Expected value: Definition and 250 Discussions

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.

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11. ### Using Multiple integrals to compute expected value

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13. M

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15. ### Show that an expected value of a vacuum state is equal to 1

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