# Several Unbiased Estimators

If I have some data set $D={X_1,...X_N}$ and have an esitmator be "pick the first point" $X_1$, how can I show that this estimator is unbiased? I also have to show why its highly undesirable, and I would assume it uses some variance/convergance argument. I would like some help proving that the estimator is indeed unbasied and in finding its variance.

• The $X$'s are independent and identically distributed, and you are trying to estimate the mean of the distribution they are drawn from?
– user14972
Oct 25 '12 at 16:48
• yeah I am trying to estimate the mean Oct 25 '12 at 17:14

If every $X_i$ has mean $\mu$ and variance $\sigma^2$, then the estimator $Y=X_1$ is such that $\mathbb E(Y)=\mu$ and $\text{var}(Y)=\sigma^2$.
Proof: $Y=X_1$. End of proof.
The fact that $\mathbb E(Y)=\mu$ may be seen as desirable if $Y$ is meant to estimate $\mu$, the fact that $\text{var}(Y)=\sigma^2$ less so if another estimator $\widetilde Y$ is such that $\mathbb E(\widetilde Y)=\mu$ and $\text{var}(\widetilde Y)\lt\sigma^2$, even less so if $\text{var}(\widetilde Y)\ll\sigma^2$.
You probably have an idea for some of these other, more efficient, estimators $\widetilde Y$.