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Hello friends!

Given an estimator of the population mean:

[tex]\bar{Y}=\frac{\sum^{N}_{i=1}Y_{i}}{N}[/tex]

The expected value of [tex]\bar{Y}[/tex] is :

[tex]E(\bar{Y}) = \frac{1}{N}E(Y_{1})+\frac{1}{N}E(Y_{2})+\cdots+\frac{1}{N}E(Y_{N})=\mu[/tex] where [tex]\mu[/tex] is the population mean.

Therefore:

[tex]E(\bar{Y}) = \frac{1}{N}\mu+\frac{1}{N}\mu+\cdots+\frac{1}{N}\mu[/tex]

My question is, why are [tex]E(Y_{1}), E(Y_{2}), E(Y_{N})[/tex] all equal to [tex]\mu[/tex]?

Given an estimator of the population mean:

[tex]\bar{Y}=\frac{\sum^{N}_{i=1}Y_{i}}{N}[/tex]

The expected value of [tex]\bar{Y}[/tex] is :

[tex]E(\bar{Y}) = \frac{1}{N}E(Y_{1})+\frac{1}{N}E(Y_{2})+\cdots+\frac{1}{N}E(Y_{N})=\mu[/tex] where [tex]\mu[/tex] is the population mean.

Therefore:

[tex]E(\bar{Y}) = \frac{1}{N}\mu+\frac{1}{N}\mu+\cdots+\frac{1}{N}\mu[/tex]

My question is, why are [tex]E(Y_{1}), E(Y_{2}), E(Y_{N})[/tex] all equal to [tex]\mu[/tex]?

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