# Thread: Show that the sample mean is an unbiased estimator of the population

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

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

2. 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.

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

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

Therefore:

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

etc...

CB

3. 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

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

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# sample mean is an unbiased estimator of population mean/derivation

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