I am trying to solve an old exercise from a statistics exam but I don't know the subject well and I hit a wall. You are given the following distribution:
\begin{align*} f(x)=\frac{2g+1}{2g}x^\frac{1}{2g} \end{align*}
With $g$ positive and $0\le x\le 1$.
I've already found an estimator for $g$, however now I am asked to find an unbiased estimator for its mean $\mu$ and prove that it is unbiased. I don't know how to do that, at least not for this particular distribution. I already found the value of $\mu$ by integrating (I found it to be $(2g+1)/(4g+1)$), and I know I somehow have to prove that the average of the estimator is equal to μ but idk how to do that. Can anyone help?
The simply way to answer is this.
An unbiased estimator for the $\mu$, the mean of the population is the sample mean
$$\hat{\mu}=\bar{X}_n$$
To prove it is unbiased is very easy
$$\mathbb{E}[\frac{1}{n}\sum_i X_i]=\frac{1}{n}\sum_i \mathbb{E}[X_1]=\frac{1}{n}\cdot n \mu=\mu$$