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GATE 2025 ST – Question 40

Standard Univariate Distributions · Continuous Distributions · 2 marks · Multiple choice

X>0 has lognormal density $f(x)=(\sigma x\sqrt{2\pi})^{-1}\exp[-\tfrac12((\ln x-\mu)/\sigma)^2]$. If ln(E(X²)/(E X)²)=4, find Var(ln X).

  1. 2
  2. 4
  3. 16
  4. 64

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Correct answer: (B) 4

Explanation

ln X is N(μ,σ²), and lognormal moments give E X=e^(μ+σ²/2),E X²=e^(2μ+2σ²). Their ratio is e^(σ²); its logarithm is σ²=4. Therefore **Var(ln X)=4, B**.

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