Statistics Question - Expected value of an estimator

[michonamona]

Hello friends!

Given an estimator of the population mean:

$$\bar{Y}=\frac{\sum^{N}_{i=1}Y_{i}}{N}$$

The expected value of $$\bar{Y}$$ is :

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

Therefore:

$$E(\bar{Y}) = \frac{1}{N}\mu+\frac{1}{N}\mu+\cdots+\frac{1}{N}\mu$$


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

 

[jbunniii]

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

Isn't that exactly what is meant by "$\mu$ is the population mean"?

By the way, you are missing a factor of 1/N in the definition of the estimator.

 

[michonamona]

Isn't that exactly what is meant by "$\mu$ is the population mean"?

By the way, you are missing a factor of 1/N in the definition of the estimator.

Thank you for your reply.

My mistake, I edited the equation.

But that's exactly what my question is about. If the Expected value of Ybar is equal to mu, then why is the expected value of EACH of the components of the series of Ybar also mu?

It must be something really simple that I'm missing...

Thanks
M

 

[jbunniii]

Thank you for your reply.

My mistake, I edited the equation.

But that's exactly what my question is about. If the Expected value of Ybar is equal to mu, then why is the expected value of EACH of the components of the series of Ybar also mu?

It must be something really simple that I'm missing...

Thanks
M

Well, what ARE these components $Y_i$? I assume they are random samples from the population, are they not?

I think you are arguing in reverse. The expected value of the estimator is $\mu$ BECAUSE the expected value of each of the random samples is $\mu$, not the other way around.

 

[michonamona]

...expected value of each of the random samples is $\mu$, not the other way around.

so each of the random sample Y_i was taken from the population? meaning the size of each Y_i is the same as the population?


Thank you,
M

 

[jbunniii]

so each of the random sample Y_i was taken from the population?

I don't know. I assume so, but you're the one who asked the original question. Is it homework? If so, doesn't the homework problem tell you what the $Y_i$'s are?

meaning the size of each Y_i is the same as the population?

I don't know what you mean by "size." If each $Y_i$ comes from the same population (more formally, the same probability distribution) then the STATISTICS should be the same for each $Y_i$. The mean ($\mu$) is one such statistic.