GATE 2026 ST – Question 40
On Ω let the σ-algebra be {Ω,∅,A,Aᶜ}, with P(A)=P(Aᶜ)=1/2 and both sets nonempty. Let X=1_A,Y=1_(Aᶜ). Which holds?
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Correct answer: (A) Same distribution function
Explanation
Both are Bernoulli(1/2), hence have the same CDF. But X+Y=1 and XY=0 everywhere, so they differ everywhere and are not independent: P(X=1,Y=1)=0≠1/4. **A**.