Question

Can anyone help me answer this?

Define what is an unbiased estimator. Show that (X ̅ ) is an unbiased estimator for E(X ̅ )=μ under the usual assumptions.

Answer 1

An unbiased estimator of a population parameter is a statistic  which represents the sample from the population and its expected value gives the true value of the population parameter.

Let x1, x2.......xn are n observations from the sample and the sample statistic which estimates population mean $\mu$ is

$\overline{X}=[ x1+x2+.....xn]/n$

If $E(\overline{X})= \mu$ then $\overline{X}$ is an unbiased estimate of population mean.

ELY ) = (1/n) * E(rl) + E(22) + El.r n

Given each observation is normally distributed around mean $\mu$

$E({\overline{X})=[ \mu +.....+\mu]/n$

$E({\overline{X})=n* \mu /n=\mu$

Hence proved