Expected value

===Infinite expected values===
Expected values as defined above are automatically finite numbers. However, in many cases it is fundamental to be able to consider expected values of {{math|±∞}}. This is intuitive, for example, in the  case of the [[St. Petersburg paradox]], in which one considers a random variable with possible outcomes {{math|''x''<sub>''i''</sub> {{=}} 2<sup>''i''</sup>}}, with associated probabilities {{math|''p''<sub>''i''</sub> {{=}} 2<sup>−''i''</sup>}}, for {{mvar|i}} ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has
<math display="block"> \operatorname{E}[X]= \sum_{i=1}^\infty x_i\,p_i  =2\cdot \frac{1}{2}+4\cdot\frac{1}{4} + 8\cdot\frac{1}{8}+ 16\cdot\frac{1}{16}+ \cdots = 1 + 1 + 1 + 1 + \cdots.</math>
It is natural to say that the expected value equals {{math|+∞}}.

There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral.{{sfnm|1a1=Billingsley|1y=1995|1loc=Section 15}} The first fundamental observation is that, whichever of the above definitions are followed, any ''nonnegative'' random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then the expected value can be defined as {{math|+∞}}. The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable {{mvar|X}}, one defines the [[positive and negative parts]] by {{math|''X''<sup> +</sup> {{=}} max(''X'', 0)}} and {{math|''X''<sup> −</sup> {{=}} −min(''X'', 0)}}. These are nonnegative random variables, and it can be directly checked that {{math|''X'' {{=}} ''X''<sup> +</sup> − ''X''<sup> −</sup>}}. Since {{math|E[''X''<sup> +</sup>]}} and {{math|E[''X''<sup> −</sup>]}} are both then defined as either nonnegative numbers or {{math|+∞}}, it is then natural to define:
<math display="block">
\operatorname{E}[X] = \begin{cases} \operatorname{E}[X^+] - \operatorname{E}[X^-] & \text{if } \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
+\infty  & \text{if } \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
-\infty & \text{if } \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] = \infty;\\
\text{undefined} & \text{if } \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] = \infty.
\end{cases}
</math>

According to this definition, {{math|E[''X'']}} exists and is finite if and only if {{math|E[''X''<sup> +</sup>]}} and {{math|E[''X''<sup> −</sup>]}} are both finite. Due to the formula {{math|{{!}}''X''{{!}} {{=}} ''X''<sup> +</sup> + ''X''<sup> −</sup>}}, this is the case if and only if {{math|E{{!}}''X''{{!}}}} is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations.
* In the case of the St. Petersburg paradox, one has {{math|''X''<sup> −</sup> {{=}} 0}} and so {{math|E[''X''] {{=}} +∞}} as desired.
* Suppose the random variable {{mvar|X}} takes values {{math|1, −2,3, −4, ...}} with respective probabilities {{math|6π<sup>−2</sup>, 6(2π)<sup>−2</sup>, 6(3π)<sup>−2</sup>, 6(4π)<sup>−2</sup>, ...}}. Then it follows that {{math|''X''<sup> +</sup>}} takes value {{math|2''k''−1}} with probability {{math|6((2''k''−1)π)<sup>−2</sup>}} for each positive integer {{mvar|k}}, and takes value {{math|0}} with remaining probability. Similarly, {{math|''X''<sup> −</sup>}} takes value {{math|2''k''}} with probability {{math|6(2''k''π)<sup>−2</sup>}}  for each positive integer {{mvar|k}} and takes value {{math|0}} with remaining probability. Using the definition for non-negative random variables, one can show that both {{math|E[''X''<sup> +</sup>] {{=}} ∞}} and {{math|E[''X''<sup> −</sup>] {{=}} ∞}} (see [[harmonic series (mathematics)|Harmonic series]]). Hence, in this case the expectation of {{mvar|X}} is undefined.
* Similarly, the Cauchy distribution, as discussed above, has undefined expectation.