Show that the sample mean is an unbiased estimator of the population

**Caption12** · April 7th 2011, 12:35 AM

Show that the sample mean is an unbiased estimator of the population mean.

i started a nw topiuc and i have been askd this. i know definition of sample mean and bias. i know unbias estimater but idont know how to show it i dont know what i plug in from puppulation mean etc

**CaptainBlack** · April 7th 2011, 01:56 AM

Originally Posted by Caption12

Show that the sample mean is an unbiased estimator of the population mean.

i started a nw topiuc and i have been askd this. i know definition of sample mean and bias. i know unbias estimater but idont know how to show it i dont know what i plug in from puppulation mean etc

You need to show that the expectation of the sample mean is equal to the population mean.

$\displaystyle \overline{x}=\frac{1}{n}\sum_{i=1}^n x_i$

where the $$$x_i$ are iid with pdf $f_X(x_i)$

Therefore:

$\displaystyle E(\overline{x})=E\left(\frac{1}{n}\sum_{i=1}^n x_i \right)=\frac{1}{n}\sum_{i=1}^n E(x_i)$

etc...

CB

**Caption12** · April 7th 2011, 02:03 AM

consider the unbias estimaters mu1hat = 1/2 xbar + 1/2 ybar. and mu2hates = 1/3 xbar + 2/3 ybar. calculate the variance of mu1hat and mu2hats

**matheagle** · April 10th 2011, 02:11 PM

we're back to the moo haters?
I mean mu haters.

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