GATE 2022 ST – Question 31

Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem · 1 mark · Numerical answer

X_i are iid with density x^(−5/6)e^(−x/8)/[√2 Γ(1/6)] on x>0. Let X̄_n be their mean. If √n[X̄_n(3−X̄_n)−20/9]→N(0,σ²) in distribution,find σ² to two decimals.

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Correct answer: 1.18 to 1.19

Explanation

The density is Gamma(shape 1/6,scale 8),so μ=4/3 and variance=32/3. For g(x)=x(3−x),g(μ)=20/9 and g′(μ)=3−8/3=1/3. The CLT and delta method give σ²=(1/3)²(32/3)=32/27≈**1.19**.

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