GATE 2020 ST – Question 33
X₁,…,Xn have density e^(−(x−2θ)) for x>2θ, zero otherwise, with θ>0 and n≥2. Let X(1)=min Xi. Find $E[\theta^{-1}(X_{(1)}-1/n)\mid X_1-X_2=2]$.
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Correct answer: 2 to 2
Explanation
Write Xi=2θ+Zi with independent rate1 exponentials. Their minimum is exponential rate n, so E X(1)=2θ+1/n. By exponential memorylessness the sample minimum is independent of all residual spacings, including X₁−X₂. Conditioning on that difference leaves the expectation unchanged. Therefore the requested value is(2θ+1/n−1/n)/θ=2. The conditional statement uses the regular conditional distribution of this continuous difference.