Expected value

===Arbitrary real-valued random variables===
All definitions of the expected value may be expressed in the language of [[measure theory]]. In general, if {{mvar|X}} is a real-valued [[random variable]] defined on a [[probability space]] {{math|(Ω, Σ, P)}}, then the expected value of {{mvar|X}}, denoted by {{math|E[''X'']}}, is defined as the [[Lebesgue integration|Lebesgue integral]]{{sfnm|1a1=Billingsley|1y=1995|1p=273}}
:<math>\operatorname{E} [X]  = \int_\Omega X\,d\operatorname{P}.</math>
Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of {{mvar|X}} is defined via weighted averages of ''approximations'' of {{mvar|X}} which take on finitely many values.{{sfnm|1a1=Billingsley|1y=1995|1loc=Section 15}} Moreover, if given a random variable with finitely or countably many possible values, the Lebesgue theory of expectation is identical with the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable {{mvar|X}} is said to be ''absolutely continuous'' if any of the following conditions are satisfied:
* there is a nonnegative [[measurable function]] {{mvar|f}} on the real line such that
::<math>\text{P}(X\in A)=\int_A f(x)\,dx,</math>
:for any [[Borel set]] {{mvar|A}}, in which the integral is Lebesgue.
* the [[cumulative distribution function]] of {{mvar|X}} is [[absolutely continuous]].
* for any Borel set {{mvar|A}} of real numbers with [[Lebesgue measure]] equal to zero, the probability of {{mvar|X}} being valued in {{mvar|A}} is also equal to zero
* for any positive number {{math|ε}} there is a positive number {{math|δ}} such that: if {{mvar|A}} is a Borel set with Lebesgue measure less than {{math|δ}}, then the probability of {{mvar|X}} being valued in {{mvar|A}} is less than {{math|ε}}.
These conditions are all equivalent, although this is nontrivial to establish.{{sfnm|1a1=Billingsley|1y=1995|1loc=Theorems 31.7 and 31.8 and p. 422}} In this definition, {{mvar|f}} is called the ''probability density function'' of {{mvar|X}} (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration,{{sfnm|1a1=Billingsley|1y=1995|1loc=Theorem 16.13}} combined with the [[law of the unconscious statistician]],{{sfnm|1a1=Billingsley|1y=1995|1loc=Theorem 16.11}} it follows that
:<math>\operatorname{E}[X]\equiv\int_\Omega X\,d\operatorname{P}=\int_{\mathbb{R}}xf(x)\,dx</math>
for any absolutely continuous random variable {{mvar|X}}. The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable.

[[File:Roland Uhl 2023 Charakterisierung des Erwartungswertes Bild1.svg|250px|right|border]]
The expected value of any real-valued random variable <math>X</math> can also be defined on the graph of its [[cumulative distribution function]] <math>F</math> by a nearby equality of areas. In fact, <math>\operatorname{E}[X] = \mu</math> with a real number <math>\mu</math> if and only if the two surfaces in the <math>x</math>-<math>y</math>-plane, described by
:<math>
x\le\mu,\;\, 0\le y\le F(x) \quad</math> or <math>\quad x\ge\mu,\;\, F(x)\le y\le 1
</math>
respectively, have the same finite area, i.e. if
:<math>
\int_{-\infty}^\mu F(x)\,dx = \int_\mu^\infty \big(1 - F(x)\big)\,dx
</math>
and both [[improper integral|improper Riemann integrals]] converge. Finally, this is equivalent to the representation
:<math>
\operatorname{E}[X] 
= \int_0^\infty \big(1 - F(x)\big)\,dx - \int_{-\infty}^0 F(x)\,dx,
</math>
also with convergent integrals.<ref>{{cite book |last1=Uhl |first1=Roland |title=Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion |date=2023 |publisher=Technische Hochschule Brandenburg |doi=10.25933/opus4-2986 |doi-access=free}} pp. 2–4.</ref>