X_n are iid Exp(rate 1). Let Y_n=|X_(2n)−X_(2n−1)| and Ȳ_n=(Y₁+⋯+Y_n)/n. If √n[exp(−Ȳ_n)−e⁻¹] converges in distribution to N(0,σ²),find σ² to two decimals.
For y≥0,P(|X₁−X₂|>y)=2P(X₁>X₂+y)=e^(−y). Thus Y_n are iid Exp(1),mean and variance 1. The CLT and delta method with g(y)=e^(−y),g′(1)=−e⁻¹ give σ²=[g′(1)]²=e⁻²≈**0.14**.
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