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GATE 2024 ST – Question 42

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

The X_i are iid N(μ,σ²). I:if [√5/(σ(2n+1))]Σᵢ₌₁ⁿi(X_i−μ) is N(0,1), then n=2. II:E{[−ln Φ((X₁−μ)/σ)]³}=6. Which hold?

  1. Only I
  2. Only II
  3. Both I and II
  4. Neither I nor II

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Correct answer: (C) Both I and II

Explanation

The variance in I is 5Σi²/(2n+1)²=5n(n+1)/[6(2n+1)]. Equating to 1 gives (5n+3)(n−2)=0, so positive integer n=2. In II, Φ of the standardized normal is uniform(0,1), and −ln U is exponential of mean 1, whose third moment is 3!=6. Both true, **C**.

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