# Asymptotically unbiasedness of estimator

I am reffering to the formula given at 2:18 in following video

Can someone explain how he arrived from this:

$\bar{X}=\frac{1}{N-1}\sum_{i=1}^Nx_{i}$

to:

$E(\bar{X})=\frac{1}{N-1}\sum_{i=1}^NE(x_{i})$

The first formula is the average of values of $x_i$. The second formula should be the "average of averages", right? If $N$ is the sample size, how can second formula be true?

• Linearity of expectation. – Christoph Hanck Dec 25 '16 at 19:53
• @ChristophHanck Could you please explain. – Quirik Dec 25 '16 at 23:25
• See your own comment below, I think you got it yourself already. – Christoph Hanck Dec 25 '16 at 23:30

• $N$ would be sample size? What do you mean when you say that $x_i$ should be considered as i.i.d. random values but not as actual sample values? – Quirik Dec 25 '16 at 22:09
• You make many assumptions that are not necessary for the equation in the question to be true, Michael: the only reason to insist on $1/N$ instead of $1/(N-1)$ is your assumption of an unstated context and that the bar has a conventional meaning; there is no need for the $X_i$ to be independent; and they needn't have a common distribution. The only necessary assumption is that all those expectations actually exist. – whuber Dec 25 '16 at 23:41