In [1, p. 60 exercise 3.4] we are asked to prove that the estimate $\hat{\mu}_x = \frac{1}{N}\sum\limits^{N-1}_{n=0}x[n]$ (1)

is an unbiased estimator, given $\left\{x,x,...,x[N]\right\}$ are independent and identically distributed according to a $N(\mu_x,\sigma^{2}_x)$ distribution. Furthermore we are asked to also find the variance of the estimator.
Solution: The mean of the estimator $\hat{\mu}_x$ is given by $E\left\{\hat{\mu}_x\right\}$ $=$ $\frac{1}{N}\sum\limits^{N-1}_{n=0}E\left\{x[n]\right\}$ $=$ $\frac{1}{N}\sum\limits^{N-1}_{n=0}\mu_x$ $=$ $\frac{1}{N} \cdot N \cdot \mu_x$ $=$ $\mu_x$

Thus the estimator $\hat{\mu}_x$ is unbiased. The variance of the estimator about the mean $\mu_x$ is given by: $E\left\{\left(\hat{\mu}_x-\mu_x\right)^2\right\}$ $=$ $E\left\{\hat{\mu}^2_x-2\hat{\mu}_x\mu_x+\mu^2_x\right\}$ $=$ $E\left\{\hat{\mu}^2_x\right\}-\mu^2_x$ $=$ $E\left\{(\frac{1}{N}\sum\limits^{N-1}_{n=0}x[n])(\frac{1}{N}\sum\limits^{N-1}_{k=0}x[k])\right\}-\mu^2_x$ $=$ $\frac{1}{N^2}\sum\limits^{N-1}_{n=0}\sum\limits^{N-1}_{k=0}E\{x[n]x[k]\} -\mu^2_x$ (2)

Considering the fact that the samples are independent we can set $E\{x[n]x[k]\}=E\{x[n]\}E\{x[k]\}=\mu^2_x$ for $n\neq k$ and obtain $\sum\limits^{N-1}_{n=0}\sum\limits^{N-1}_{k=0}E\{x[n]x[k]\}$ $=$ $\sum\limits^{N-1}_{n=0}E\{x^2[n]\}$  $+{\sum\limits^{N-1}_{n=0, n \neq k}\sum\limits^{N-1}_{k=0}}E\{x[n]\}E\{x[k]\}$ $=$ $N \cdot E\{x^2[n]\} +N(N-1)\mu^2_x$ (3)

Thus it remains to obain the value of $E\{x^2[n]\}$ in order to determine the variance of the estimator. We can determine $E\{x^2[n]\}$ by the variance $\sigma^2_x$ of $x[n]$: $\sigma^2_x$ $=$ $E\{(x[n]-\mu_x)^2\}$ $=$ $E\{x^2[n]\}-\mu^2_x \Rightarrow$ $E\{x^2[n]\}$ $=$ $\sigma^2_x+\mu^2_x$ (4)

And thus the variance of the estimator (2), considering (3) and (4), is given by: $E\left\{\left(\hat{\mu}_x-\mu_x\right)^2\right\}$ $=$ $\frac{1}{N} (\sigma^2_x+\mu^2_x+(N-1)\mu^2_x) -\mu^2_x$ $=$ $\frac{1}{N} \sigma^2_x.$ (5)

We have proved thus that the estimator is unbiased and have obtained the variance of the estimator $\hat{\mu_x}$. QED.

 Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.