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GATE 2021 ST – Question 44

Joint Distributions · Joint and Conditional Distributions · 2 marks · Multiple choice

The joint density is 4/(x+y)³ for x>1,y>1,zero elsewhere. Which claim is NOT true?

  1. The density of X+Y is 4(z−2)/z³ for z>2.
  2. P(X+Y>4)=3/4.
  3. E(X+Y)=4ln 2.
  4. E(Y | X=2)=4.

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Correct answer: (C) E(X+Y)=4ln 2.

Explanation

For sum z>2,the integration interval 1<x<z−1 has length z−2,giving A. Its survival at 4 is ∫₄∞4(z−2)/z³ dz=3/4. Its mean integral has tail proportional to ∫dz/z and diverges,so C is false. The marginal f_X(x)=2/(x+1)² and conditional integration give E(Y|X=x)=x+2,hence D. **C**.

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