conditional expectation of gamma distribution with $\alpha = 1$

Question

$X_i$ are exponential($\lambda$) distribution and identically independent distribution.

$Y = \sum_{i=1}^nX_i$

$X_i$ is an unbiased estimator of $\lambda$.

$Y$ is a sufficient estimator of $\lambda$.

solve $E[X_1|Y]$

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I know that $Y$ is Gamma distribution // $gamma(1,1/\lambda)$

but i can't solve this problem.

how to solve this problem?

Answer 1

Hint:

Consider $$\mathbb E\left[X_1 \mid Y\right] = E\left[X_j \mid Y\right] = \frac1n\sum_{i=1}^n \mathbb E\left[X_i \mid Y\right]= \frac1n\mathbb E\left[\sum_{i=1}^nX_i \;\Bigg| \; Y\right]$$