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

Why are the following estimators of \mu unsatisfactory?

i) X_1

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

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

Thanks any help would be appreciated.