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I don't really know how to start proving this question.

Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite.

Show that $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$

Does anyone here have any idea for starting this question?

  • 2
    As this question has no explicit mention of stochastic processes, I replaced the tag with more appropriate ones.2011-11-03
  • 0
    [Thomas Andrews' answer](http://math.stackexchange.com/questions/139405/conditional-expectation-of-book-shiryaev-page-233/139407#139407) to a re-posting of essentially this same question is simple and elegant.2012-05-01
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    This question has been re-asked at least twice (it's a popular textbook exercise), so I edited the title in hopes of making it easier to find. If someone can improve it further, please do.2013-05-31
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    See also an answer [here](https://math.stackexchange.com/questions/1842364/conditional-expectation-of-independent-variables/1842770#1842770).2017-08-09

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