The GATE Grind

GATE 2023 ST – Question 18

Standard Univariate Distributions · Continuous Distributions · 1 mark · Multiple choice

X has gamma density λ^α x^(α−1)e^(−λx)/Γ(α),x>0,α,λ>0. Which claim is NOT true?

  1. E X exists for all positive α,λ.
  2. Var X exists for all positive α,λ.
  3. E(1/X) exists for all positive α,λ.
  4. E[ln(1+X)] exists for all positive α,λ.

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Correct answer: (C) E(1/X) exists for all positive α,λ.

Explanation

Gamma mean and variance are α/λ and α/λ². Since 0≤ln(1+x)≤x,the logarithmic expectation is finite. Near zero,the inverse-moment integrand is proportional to x^(α−2),integrable only if α>1. Thus **C** fails for 0<α≤1.

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