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Let $X_1$ and $X_2$ be i.i.d. random variables, and define $S_2 = X_1 + X_2$. Then $$\mathrm{E}[X_1|S_2] = \frac{X_1+X_2}2=\frac{S_2}2.$$ I don't understand why the conditional expectation is the average of the X's. Can anyone help me with that?

Sandejo
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Vic Anderson
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  • https://math.stackexchange.com/questions/78546/conditional-expectation-for-a-sum-of-iid-random-variables-e-xi-mid-xi-eta-e?noredirect=1&lq=1 – StubbornAtom Oct 18 '21 at 09:55

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Since $X_1=S_2-X_2$, we can rewrite the conditional expectation as $$\mathrm{E}[X_1|S_2] = \mathrm{E}[S_2-X_2|S_2] = S_2-\mathrm{E}[X_2|S_2].$$ Since $X_1$ and $X_2$ are i.i.d, $\mathrm{E}[X_1|S_2]=\mathrm{E}[X_2|S_2]$. Therefore, the equation above simplifies to $$2\mathrm{E}[X_1|S_2] = S_2.$$

Sandejo
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