$X_{1},X_{2},...X_{n}$ i.i.d. ~ $N(\mu,\theta)$,calculate $E[X_{1}|\bar X]$

Question

$X_{1},X_{2},...X_{n}$ i.i.d. ~ $N(\mu,\theta)$, I know $E(X_{1}|\bar X)$ is UMVUE, but how can I calculate $E[X_{1}|\bar X]$, should I find the joint distribution of $X_{1}\&\bar X$? I think there is other way to do it, I have idea for continuous case, dose anyone could help me? Thanks a lot.

Answer 1

On the one hand, $$ \mathbb E\left[\sum_{i=1}^{n}X_i\bigg\vert \bar{X}\right]=\mathbb E\left[n\bar{X}\vert \bar{X}\right]=n\mathbb E\left[\bar{X}\vert \bar{X}\right]=n\bar{X}\tag{1} $$

while, on the other hand, $$ \mathbb E\left[\sum_{i=1}^{n}X_i\bigg\vert \bar{X}\right]= \sum_{i=1}^{n}\mathbb E\left[X_i\vert \bar{X}\right]=n\mathbb E\left[X_1\vert \bar{X}\right]\,.\tag{2} $$

Therefore,

$$ \mathbb E\left[X_1\vert \bar{X}\right]=\bar{X} $$