Let $X_1, X_2, ..., X_n$ be a random iid sample from a population with mean $\theta$.
Now I am wondering about the intuition behind $E(X_1| \overline X ) = \overline X$, the sample mean.
If we just consider $X_1$ (or any $X_i$ for that matter) we have that $E(X_1) = \theta$ as the expected value of any random observation from a population will be the population mean.
Now given that we know $\overline X$ how come that changes what we expect to get for $X_1$? $X_1$ is still the same random observation from the population as before...but it seems that knowing the sample mean 'overrides' what we expect to get for an observation...is this correct?