Given a random variable $x$ with a well-defined expected value $\mu$, is the mean of the set of samples $\{x_1,\ \cdots,\ x_n\}$, which we'll call $\widehat{\mu}$, always an unbiased estimator of $\mu$? In other words, is it always true that:
$$
E[\widehat{\mu}] = E[\frac{\sum_{i=1}^n x_i}{n}] = \mu = E[x].
$$
regardless of the specifics of $x$?
Further, would $\widehat{\mu}$ always be consistent, in the sense that the variance $V[\widehat{\mu}]$ would tend to decrease as the sample count $n$ increased?
Best Answer
Answered in comments: The first question is answered immediately using the linearity of expectation. The second conclusion is true only when the underlying distribution has finite variance, in which case it follows with a simple computation of the variance. – whuber
The second conclusion even follows without assuming finite variance, since you assumed the mean $\mu$ exists. The strong law of large numbers then give the result, it can be proved without assuming finite variance.