Unbiased estimators

**chella182** · August 3rd 2009, 05:50 AM

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 $X_1$ , $X_2$ ,..., $X_n$ are a random sample from a population with mean $\mu$ and variance $\sigma^2$ .
a) Prove that the mean estimator $\bar{X}$ is unbiased for $\mu$ . State carefully any formulae you use.
b) Prove that $\bar{X}$ has variance $\frac{\sigma^2}{n}$ . 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.

**CaptainBlack** · August 3rd 2009, 06:13 AM

Originally Posted by chella182

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 $X_1$ , $X_2$ ,..., $X_n$ are a random sample from a population with mean $\mu$ and variance $\sigma^2$ .
a) Prove that the mean estimator $\bar{X}$ is unbiased for $\mu$ . State carefully any formulae you use.
b) Prove that $\bar{X}$ has variance $\frac{\sigma^2}{n}$ . 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.

$\overline{X}=\frac{1}{n}\sum_{i=1}^n X_i$

By linearity of the expectation operator:

$E(\overline{X})=\frac{1}{n}\sum_{i=1}^n E(X_i)$

but by definition $E(X_i)=\mu$ , so:

$E(\overline{X})=\frac{1}{n}\sum_{i=1}^n \mu=\mu$

Which is the definition of an unbiased estimator for $\mu$ .

CB

**chella182** · August 3rd 2009, 06:16 AM

Thank you. Does sound rather simple, I just find it difficult to get my head around this sort of stuff.

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