Expected value

===Random variables with density===
Now consider a random variable {{mvar|X}} which has a [[probability density function]] given by a function {{mvar|f}} on the [[real number line]]. This means that the probability of {{mvar|X}} taking on a value in any given [[open interval]] is given by the [[integral]] of {{mvar|f}} over that interval. The expectation of {{mvar|X}} is then given by the integral{{sfnm|1a1=Papoulis|1a2=Pillai|1y=2002|1loc=Section 5-3|2a1=Ross|2y=2019|2loc=Section 2.4.2}}
:<math>\operatorname{E}[X] = \int_{-\infty}^\infty x f(x)\, dx.</math>
A general and mathematically precise formulation of this definition uses [[measure theory]] and [[Lebesgue integration]], and the corresponding theory of ''absolutely continuous random variables'' is described in the next section. The density functions of many common distributions are [[piecewise continuous]], and as such the theory is often developed in this restricted setting.{{sfnm|1a1=Feller|1y=1971|1loc=Section I.2}} For such functions, it is sufficient to only consider the standard [[Riemann integration]]. Sometimes ''continuous random variables'' are defined as those corresponding to this special class of densities, although the term is used differently by various authors.

Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of {{mvar|X}} is given by the [[Cauchy distribution]] {{math|Cauchy(0, π)}}, so that {{math|''f''(''x'') {{=}} (''x''<sup>2</sup> + π<sup>2</sup>)<sup>−1</sup>}}. It is straightforward to compute in this case that
:<math>\int_a^b xf(x)\,dx=\int_a^b \frac{x}{x^2+\pi^2}\,dx=\frac{1}{2}\ln\frac{b^2+\pi^2}{a^2+\pi^2}.</math>
The limit of this expression as {{math|''a'' → −∞}} and {{math|''b'' → ∞}} does not exist: if the limits are taken so that {{math|''a'' {{=}} −''b''}}, then the limit is zero, while if the constraint {{math|2''a'' {{=}} −''b''}} is taken, then the limit is {{math|ln(2)}}.

To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral [[converges absolutely]], with {{math|E[''X'']}} left undefined otherwise.{{sfnm|1a1=Feller|1y=1971|1p=5}} However, measure-theoretic notions as given below can be used to give a systematic definition of {{math|E[''X'']}} for more general random variables {{mvar|X}}.