Project ForgeEngineering Mathematics
Probability & Statistics
GATE 2014
Hard
Consider the following two normal distributions: f1(x)=exp(−πx²), f2(x)=(1/(2π))exp{−(1/(4π))(x²+2x+1)}. If μ and σ denote the mean and standard deviation, respectively, then
- Aμ1<μ2 and σ1²<σ2²
- Bμ1<μ2 and σ1²>σ2²
- μ1>μ2 and σ1²<σ2²
- Dμ1>μ2 and σ1²>σ2²
Explanation
Matching to N(μ,σ²)∝exp(−(x−μ)²/2σ²): f1=exp(−πx²) gives μ1=0, 1/(2σ1²)=π ⟹ σ1²=1/(2π). f2∝exp(−(x+1)²/(4π)) gives μ2=−1, 1/(2σ2²)=1/(4π) ⟹ σ2²=2π. So μ1=0 > μ2=−1, and σ1²=1/(2π) < σ2²=2π.
Source: GATE CH 2014 (2014)
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