# Unbiased estimator for Theta of a Normal Distribution

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.

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