Suppose we have a random sample ($\displaystyle X_1,X_2, ... ,X_n$) drawn from a Normal random variable $\displaystyle X \sim N(\mu , \sigma^2)$.

Why are the following estimators of $\displaystyle \mu$ unsatisfactory?

i) $\displaystyle X_1$

Do I just need to prove biasedness for this? i.e. $\displaystyle E(X_1) = X_1 \not= \mu $?

Also I have another question ... is $\displaystyle \bar{x} = \mu $ i.e. $\displaystyle E(\bar{x}^2) = E(\mu^2) = \frac{\sigma^2}{n} + \mu^2$ ?

Thanks any help would be appreciated.