GATE 2026 ST – Question 63
Independent U₁,U₂,U₃ are exponential with respective rates 1,2,3. Let X=min(U₁,U₃),Y=min(U₂,U₃). Find P(X=Y) to two decimals.
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Correct answer: 0.47 to 0.53
Explanation
Continuous independent arrival times have no ties. X and Y are equal precisely when their common variable U₃ is the earliest of all three. Competing exponential rates give probability 3/(1+2+3)=**0.50**; direct integration is ∫₀∞3e⁻³ᵗe⁻ᵗe⁻²ᵗdt=1/2.