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Let X be a random variable. Will its expectation E[X] expressed as a (possibly infinite) sum converge for arbitrary X?

My intuition would be yes. Since E[X] is a function of a r.v., E[X] is a r.v. itself, so by definition it maps to the reals. The infinity is not a real number, so there will be a limit to which the E[X] will converge.

Of course you can construe a r.v. such that weighted sum will diverge, but is then the case that E[X] = $\infty$ or is E[X] said to be undefined?

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  • $\begingroup$ See stats.stackexchange.com/questions/36027/… $\endgroup$ – Tim Nov 13 '17 at 12:55
  • $\begingroup$ Is this example enough for you? $\endgroup$ – Tim Nov 13 '17 at 13:40
  • $\begingroup$ Yes, it was more or less a question about the convention and it seems that by convention the divergent series results in an undefined expectation. $\endgroup$ – martn Nov 13 '17 at 14:55
  • $\begingroup$ OK, so I'm marking it as a duplicate. If you'd feel that there is something more in your question that needs answering, please edit to make it more clear and it could be re-opened. The idea of duplicates is that they may it easier for others to find relevant threads and it helps us not repeating similar Q&A's. $\endgroup$ – Tim Nov 13 '17 at 14:57