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I have a problem that can be resolved if i show that $E(\varepsilon_k\mid\sigma(\varepsilon_1,\ldots,\varepsilon_{k-1}))=E(\varepsilon_k)$ where $\varepsilon_1,\ldots,\varepsilon_k$ $\sim \mathcal{N}(0,1)$ and i know they are independent.

I dont know where to even start. Any proof or help would be great.

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    Davide Giraudo: Is that not what i want to show? @NateEldredge: The Definition of conditional expectation is the real-valued random variable satisfying that $ \int_D X dP = \int_D E(X| \mathbb{D}) dP $ for every $D \in \mathbb{D}$. Can you give me a Hint on how to use that?2012-12-16

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By the definition of conditional expectation, it suffices to show $\int_D \varepsilon_k \,dP = \int_D E[\varepsilon_k]\,dP$ for every $D \in \sigma(\varepsilon_1, \dots, \varepsilon_{k-1})$. In other words, to show that $E[1_D \varepsilon_k] = E[1_D] E[\varepsilon_k]$. But what do you know about the random variables $\varepsilon_k$ and $1_D$?