One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation $\sigma_o$.
The $\sigma_o$ is estimated by randomly drawing out $10,000$ numbers of samples $\left(x_n\right)$. The estimates $\hat{\sigma}_1, \hat{\sigma}_2$ are computed in the following two ways.
$$\hat{\sigma}_1^2=\frac{1}{10000} \sum_{n=1}^{10000} x_n^2 \quad \hat{\sigma}_2^2=\frac{1}{9999} \sum_{n=1}^{10000} x_n^2$$
Which of the following statements is true?
- $E\left(\hat{\sigma}_2^2\right)=\sigma_o^2$
- $E\left(\hat{\sigma}_2\right)=\sigma_o$
- $E\left(\hat{\sigma}_1^2\right)=\sigma_o^2$
- $E\left(\hat{\sigma}_1\right)=E\left(\hat{\sigma}_2\right)$