Beyond expectations
Does there exist a random variable such that the probability of it exceeding or equaling its expected value is 0?
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1 solution
This is not real satisfying. "Expected values" typically have real life meaning, so what would be a real life example of this counterexample? Are there any counterexamples that do not appeal to the infinite?
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- infinite expectation are possible in reality (see @Agnishom Chattopadhyay ’s reply).
- no: there is no example of a r. v. with a non-infinite expected value, with the property in the original problem (see my post below).
oh wow...but i don't really know calculus. can you give me a non calculus explanation? thanks!!!
Poorly worded problem. Can't make simplistic remarks about equaling or exceeding infinity.
This is rubbish, one may only talk of ‘the’ expected value of a real-valued r. v., X , if and only if it is in L 1 , which requires − ∞ < E [ X ] < ∞ . Now, suppose P [ X ≥ m ] = 0 , where m : = E [ X ] and the expected value exists. Then m − X > 0 a.e. By basic measure theory, this implies E [ m − X ] > 0 . But E [ m − X ] = m − E [ X ] = 0 . This is a contradiction. Hence the assumption was wrong, and P [ X ≥ m ] = 0 .
Relevant wiki: Expected Value - Problem Solving
Consider a variable x = n 1 , where n is randomly chosen from the interval ( 0 , 1 ) . A simple integration can show that the expected value of x is ∞ , which is obviously unattainable.