So I'm revising my stats by doing a past exam paper, and I'm stuck on one of the questions about unbiased estimators. The question goes...

*Suppose that $\displaystyle X_1$, $\displaystyle X_2$,..., $\displaystyle X_n$ are a random sample from a population with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$.*

**a)** Prove that the mean estimator $\displaystyle \bar{X}$ is unbiased for $\displaystyle \mu$. State carefully any formulae you use.

**b)** Prove that $\displaystyle \bar{X}$ has variance $\displaystyle \frac{\sigma^2}{n}$. State carefully any formulae you use.

It's probably really simple like the last unbiased estimator Q I posted a while back, but I'm just stumped.