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A strange thing my professor told me about expectation

  1. Sep 28, 2011 #1
    Suppose you know [itex]E[X] = 0[/itex] for a given (continuous) random variable. Does that mean [itex]E[|X|] < \infty[/itex]? This is what my professor told me today, though it doesn't really make much sense...
  2. jcsd
  3. Sep 28, 2011 #2


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    It is a subtle point. If E(|X|) is infinite, then according to some definitions, E(X) does not exist.
    On the other hand if the distribution is Cauchy { density 1/[π(1+x2)]}, then E(X)=0 by symmetry even though E(|X|) is infinite.
  4. Oct 1, 2011 #3
    Ok, thanks. Looking back in our textbook, the definition of "expectation" begins: "Suppose [itex]\int_\Omega |X(\omega)| dP(\omega) <\infty[/itex]. Then we define the expectation..." So I guess that's the way it is!
  5. Oct 1, 2011 #4
    If X is Cauchy distributed, then E[X] doesn't exist. So saying E[X]=0 is wrong there. It's as wrong as saying [itex]\int_{-\infty}^{+\infty}{xdx}=0[/itex]. The integral simply does not exist.
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