When computing expected values, Riemann integration works for only random variables with bounded support sets. For distributions with unbounded support sets, we can use improper Riemann integrals for "nice" cases like the normal or gamma distribution; these integrals do converge.

Theoretically, I know that there are "nasty" cases that require more complicated forms of integrals, because the Riemann integrals don't converge. For these cases, we need other forms of integration, like Lebesgue integrals.

However, in real-life applications in statistics, are there any continuous distributions whose expectations involve "nasty" integrals that don't converge?

  • $\begingroup$ See, inter alia, this comment by Kjetil Halvorsen. For an abundance of examples, see any text on stochastic processes in finance. $\endgroup$ – whuber Oct 30 '20 at 19:19
  • $\begingroup$ @whuber #1) Are these distributions commonly used in industry? #2) Do practitioners in these fields regularly use Lebesgue integration to compute the expected values such random variables in these industries? $\endgroup$ – Iterator516 Oct 30 '20 at 22:16
  • $\begingroup$ Yes and yes. Actually, in statistical finance people use a generalization of Lebesgue-Stieltjes integration to compute Ito or Stratonovich integrals. $\endgroup$ – whuber Oct 31 '20 at 15:22

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