One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation 𝜎 0 . The 𝜎 0 is estimated by randomly drawing out 10,000 numbers of samples (𝑥 𝑛 ). The estimates 𝜎̂ 1 , 𝜎̂ 2 are computed in the following two ways. \sigma ^{\^}_{1}^{2} = \frac{1}{10000}\sum _{n = 1}^{10000}x_{n}^{2} \sigma ^{\^}_{2}^{2} = \frac{1}{9999}\sum _{n = 1}^{10000}x_{n}^{2}

Correct Answer is : E(σ^12)=σ02

E(\sigma ^{\^}_{2}^{2}) = \sigma _{0}^{2}

E(\sigma ^{\^}_{2}) = \sigma _{0}

E(\sigma ^{\^}_{1}^{2}) = \sigma _{0}^{2}

E(\sigma ^{\^}_{1}) = \sigma ^{\^}_{2}

GATE Electrical Engineering (EE) 2023 Solved Paper

Question : 25 of 65

Marks: +1, -0

One million random numbers are generated from a statistically stationary process with a Gaussian distribution with mean zero and standard deviation 𝜎₀.
The 𝜎₀ is estimated by randomly drawing out 10,000 numbers of samples (𝑥_𝑛). The estimates 𝜎̂₁, 𝜎̂₂ are computed in the following two ways.

σ ↖{\^}_{1}^{2} = \;{1}/{10000}Σ _{n = 1}^{10000}x_{n}^{2}

σ ↖{\^}_{2}^{2} = \;{1}/{9999}Σ _{n = 1}^{10000}x_{n}^{2}

$E(σ ↖{\^}_{2}^{2}) = σ _{0}^{2}$
$E(σ ↖{\^}_{2}) = σ _{0}$
$E(σ ↖{\^}_{1}^{2}) = σ _{0}^{2}$
$E(σ ↖{\^}_{1}) = σ ↖{\^}_{2}$

Validate

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