Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have: $$ \mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}. $$ Proof. To see this, I have proceeded as follows. From the general properties of conditional expectation, we have: $$ \mathbb E[Z_1+Z_2|Z_1+Z_2] =Z_1+Z_2. $$
Now, again from the general properties of c.e. (linearity), I can write: $$ \mathbb E[Z_1+Z_2|Z_1+Z_2] =\mathbb E[Z_1|Z_1+Z_2] +\mathbb E[Z_2|Z_1+Z_2] =2\mathbb E[Z_1|Z_1+Z_2] $$ The last equality because $Z_1$ has the same distribution as $Z_2$. Putting the two equalities together we find immediately the identity stated in the Claim.
Question What I don't find clear is where exactly the condition of independence of the two random variables is used. I know that the result is not true if they aren't independent, but I don't see where this conditoin is needed. The only point I can think of is in the last step of the chain of equalities, where we use the fact that $\mathbb E[Z_1|Z_1+Z_2]=\mathbb E[Z_2|Z_1+Z_2]$ but it seems to me that this holds just beacuse of the same distribution of $Z_1, Z_2$ and does not require independence.
And yet, the result is not true if $Z_1, Z_2$ are not independent. So, where is the independece condition used?
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