# Statistics Question - Expected value of an estimator

1. Mar 24, 2010

### 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 $$\mu$$?

Last edited: Mar 24, 2010
2. Mar 24, 2010

### jbunniii

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.

3. Mar 24, 2010

### michonamona

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

4. Mar 24, 2010

### jbunniii

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.

5. Mar 24, 2010

### michonamona

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

6. Mar 24, 2010

### jbunniii

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?

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.