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I'm working on some statistics project and am not getting further because of some stupid prediction that doesn't want to be 0. That's why I was wondering if maybe the following holds: Suppose we have a random variable X and an unknown constant m. Is then $E\left(X|X+m\right)=E(X)?$ That would help a lot. I thought it might hold because m is unknown and thus knowing X+m doesn't say anything about X, so one could consider X and X+m as being independent?

Thanks a lot!

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    Pity. Thank you guys!2012-07-02

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one could consider $X$ and $X+m$ as being independent?

Certainly not. If $m$ is independent of $X$, then $X$ and $X+m$ are positively correlated.

If $m$ is constant, then $E(X|X+m)=X$ because we are conditioning on the $\sigma$-algebra generated by $X+m$, which is the same as $\sigma$-algebra generated by $X$. If $m$ is a random variable, then one should look into the joint distribuition of $X$ and $m$ to say anything about $E(X|X+m)$.