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If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met?

I am facing difficulty in solving whenever there is multi variate Random variables, and how do we represent $\bar{X}_n$.

It would be nice if someone could help me solve it.

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$$E[\bar{X}_n] = \operatorname E\left[\frac{1}{n}\sum_{i=1}^n X_i\right] = \frac{1}{n}\sum_{i=1}^n \operatorname E[ X_i] = \frac{1}{n}\cdot n\operatorname E[ X_i] = \theta$$

I hope this is the right proof, not sure about the second part. Open to discussion, if you find anything wrong with the proof do tell me. Thank you :)

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