# Bias of functions defined on samples for population

#### mathinator

##### New member
Let X1, · · · , Xn be a simple random sample from some finite population of values {x1, · · · xN }.
Is the estimate $$\displaystyle \frac{1}{n} \sum_{i}^{n} f(Xi)$$ always unbiased for $$\displaystyle \frac{1}{N} \sum_{i}^{N} f(xi)$$ no matter what f is?

My thinking: I don't think all f's are unbiased, because not all sample parameters (ex: variance, or s^2) are unbiased for the population parameter (unless they are corrected for finite population sampling). I am confused if I am interpreting the question correctly, i.e f refers to parameters we can kind about the population

#### Jameson

Staff member
Hi mathinator,

Welcome to MHB!

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.

#### mathinator

##### New member
Hi mathinator,

Welcome to MHB!

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.