Let X be a symmetric random variable around 0 so that ∀ x ∈ R we have P ( X ⩽ − x ) = P ( X ⩾ x ). And it's clear that the expectation value E X = 0.My question is that, given any ε &gt; 0, whether or not E [ X I { | X | &lt; ε } ] = 0 still holds, where I is indicator function.Intuitively I think it's right, such as taking X ∼ N ( 0 , 1 ). But when I try to show it in general case, I have to proof that the following Lebesgue integral E [ X I { | X | &lt; ε } ] = ∫ | X | &lt; ε X d P = 0and I'm stuck here. Can you help me? Thanks a lot.

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Let   X be a symmetric random variable around 0 so that   ∀  x  ∈      R   we have       P    (  X  ⩽  −  x  )  =      P    (  X  ⩾  x  ). And it&#039;s clear that the expectation value       E    X  =  0.My question is that, given any   ε  &amp;gt;  0, whether or not       E    [  X      I          {              |            X              |            &amp;lt;      ε      }        ]  =  0 still holds, where   I is indicator function.Intuitively I think it&#039;s right, such as taking   X  ∼      N    (  0  ,  1  ). But when I try to show it in general case, I have to proof that the following Lebesgue integral  E  [  X      I          {              |            X              |            &amp;lt;      ε      }        ]  =      ∫                  |            X              |            &amp;lt;      ε        X        d        P    =  0and I&#039;m stuck here. Can you help me? Thanks a lot.

robegarj · Accepted Answer

Saying that   X is symmetric wrt 0 can also be done by stating that   X and   −  X have the same distribution.Then consequently   f  (  X  ) and   f  (  −  X  ) have the same distribution for every Borel measurable function   f  :      R    →      R  .For   f you can take the function prescribed by   x  ↦  x      1          [      0      ,      ϵ      )        (      |    x      |    ).Then we find that   X      1          [      0      ,      ϵ      )        (      |    X      |    ) and   −  X      1          [      0      ,      ϵ      )        (      |    −  X      |    )  =  −  X      1          [      0      ,      ϵ      )        (      |    X      |    ) have equal distribution so that they have equal expectations.

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