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Was anyone able to explain why $$E(U_2) = 0$$

I don't quite understand what the relevance of the underlined statement - "by the symmetry of $U_1$" in determining $E(U_2)$ is enter image description here

edit: I get it now, was brain dead on the night of asking this question. thanks for the responses anyway

elbarto
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    Could you please state the book from which this exercise is taken? Thanks. – COOLSerdash Nov 08 '14 at 18:30
  • Hint: how are the distributions of $U_2$ and $-U_2$ related? What does that imply about how their expectations are related? – whuber Nov 08 '14 at 18:37
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    The textbook is making a lot of fuss over nothing. Since $U_1$ and $U_2$ are _independent_, $E[U_2\mid U_1] = E[U_2]$ without the necessity of dragging in covariances (which might not even exist). The only issue here is why $E[U_2] = 0$ and the reason given for this, while perfectly correct in this simple situation, need not apply in general: symmetry of the density need not mean that $E[U] = 0$ as [the well-known example of Cauchy random variables](http://stats.stackexchange.com/q/36027/6633) shows. – Dilip Sarwate Nov 08 '14 at 22:52
  • @DilipSarwate I am confused as to why $U_1 \neq 0$ and $U_2$ equals zero? Aren't they just arbitrary random variables with the same distribution? – elbarto Nov 09 '14 at 02:13
  • @COOLSerdash it's from a homework question. – elbarto Nov 09 '14 at 02:28
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    @whuber $U_2$ and $-U_2$ have the same distribution? – elbarto Nov 09 '14 at 02:29
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    The statement "... $E[U_2]=0$ by symmetry of $f_{U_1}$." has the obvious typo that it should have said "$f_{U_2}$." – Dilip Sarwate Nov 09 '14 at 05:04
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    FWIW, the duplicate was found as the first relevant hit in a site search for [distribution symmetry expectation](http://stats.stackexchange.com/search?tab=relevance&q=distribution%20symmetry%20expectation). The next hit I could find is http://stats.stackexchange.com/questions/46843, which directly addresses a generalization of the present question (to all odd moments). (Most of the other hits do not discuss symmetry *per se*, but only invoke it without further comment.) – whuber Nov 09 '14 at 17:38

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