Let $X_1, ... ,X_n$ be i.i.d with a common density function: $f_x$, with finite mean $\mu$ and variance $ 0 < \sigma^2 < \infty$.
Consider the estimator $\hat \mu_n = \frac{1}{n-1} \sum_{i=1}^{n} X_i$.
Is the estimator biased or unbiased?
Update:
Given that I know it is a biased estimator now: I calculate the bias as
$E(\hat \mu_n) - \mu$ => $\frac{n\mu}{n-1} - \mu$ => $(\frac{1}{n-1})\mu$ => bias = $\frac{\mu}{n-1}$
Is this correct?
An unbiased estimator is one in which the expected value of the estimator is equal to the parameter to estimate.
This is a biased estimator for the mean $\mu$ of the distribution.
Your calculation has a mistake as sum is from 1 to n:
$E(\hat \mu_n) = \frac{1}{n-1} \sum_{i = 1}^{n} \mu = \frac{1}{n-1} (n)(\mu) \neq \mu$.
But note that the estimator is consistent as when $n \rightarrow \infty $ the estimator $\rightarrow \mu$ .
Update: Yes, you have correctly calculated the bias of your estimator $\hat \mu_n$ to be $\frac{\mu}{n-1}$.