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GATE 2024 ST – Question 20

Joint Distributions · Joint and Conditional Distributions · 1 mark · Multiple choice

Which statement is not true for random variables X,Y?

  1. Equal CDFs can occur with P(X≠Y)>0.
  2. Equal CDFs can occur with P(X=Y)=0.
  3. Independent X,Y imply independent X²,Y².
  4. Independent X²,Y² imply independent X,Y.

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Show answer and explanation

Correct answer: (D) Independent X²,Y² imply independent X,Y.

Explanation

Take X=Y to be a nondegenerate fair sign ±1. Their squares are both constant 1 and hence independent, while X,Y themselves are perfectly dependent, refuting D. Independent continuous variables with equal distributions verify A,B. Measurable functions of independent variables remain independent, verifying C. **D**.

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