GATE 2023 EE – Question 25
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 ($x_n$). The estimates $\hat\sigma_1,\hat\sigma_2$ are computed in the following two ways.
$$\hat\sigma_1^2=\frac1{10000}\sum_{n=1}^{10000}x_n^2\qquad\hat\sigma_2^2=\frac1{9999}\sum_{n=1}^{10000}x_n^2$$
Which of the following statements is true?
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Show answer and explanation
Correct answer: (C) $E(\hat\sigma_1^2)=\sigma_o^2$
Explanation
The mean is known to be zero, so $E[x_n^2]=\sigma_o^2$ and $E[\hat\sigma_1^2]=\dfrac{10000\,\sigma_o^2}{10000}=\sigma_o^2$: unbiased (C). $\hat\sigma_2^2$ divides by 9999 and so has expectation $\frac{10000}{9999}\sigma_o^2\neq\sigma_o^2$ (A false). Square roots of unbiased variance estimates are biased (B false), and $\hat\sigma_1\ne\hat\sigma_2$ in expectation (D false).