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Distribution with no fractional moments?

  1. Feb 21, 2013 #1
    Is it possible that for a random variable X there exist no p in (0,1) so that EX^p < ∞ ?
    Is there any example?
  2. jcsd
  3. Feb 23, 2013 #2
    Yes, many examples can be constructed by applying a sufficiently rapidly growing function to one of the standard random variables, e.g. exp(N^2) where N has the geometric distribution.
  4. Feb 23, 2013 #3


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    Well, the nearest I readily remember is Cauchy distbn, the case of p=1 (which of course is excluded by your condition). The condition 0<p<1 restricts us to only non negative variables.
  5. Feb 23, 2013 #4
    I actually meant E|X|^p < ∞ to include RV's on R
  6. Feb 23, 2013 #5
    Try to find one yourself.

    Let your phase space be [0,1] with the usual Borel structure and the usual measure.

    Define the random variable [itex]X:[0,1]\rightarrow \mathbb{R}:x\rightarrow x^{-1/n}[/itex]. What can you say about [itex]E[|X|^p][/itex]? For which p does this exist?

    Can you find a function on [itex][0,1][/itex] which is larger then any [itex]x^{-1/n}[/itex] as soon as x is close to 0?? What about the p'th moments of those functions?
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