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GATE 2026 ST – Question 53

Convergence of Random Variables · Modes of Convergence · 2 marks · Multiple select

For n≥1, P(X_n=x)=[1/(5n)](1−1/(5n))^x,x=0,1,…. Let Z_n=X_n/n converge in distribution to V. Which hold?

  1. V is N(5,25)
  2. V is χ² with 5 degrees of freedom
  3. V is exponential of mean 5
  4. e^(−V/5) is uniform on (0,1)

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Correct answer: (C) V is exponential of mean 5; (D) e^(−V/5) is uniform on (0,1)

Explanation

For z≥0, P(Z_n>z)=(1−1/(5n))^(floor(nz)+1)→e^(−z/5). Thus V is exponential of mean 5. For 0<u<1, P(e^(−V/5)≤u)=P(V≥−5ln u)=u, proving the uniform transform. **C,D**.

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