GATE 2020 ST – Question 63

Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem · 2 marks · Numerical answer

Iid Xᵢ have EX₁=0,EX₁²=1,EX₁⁴=3. Let $Y_n=n^{-1}\sum X_i^2$. If $\lim P(Y_n+\sqrt n(Y_n-1)/\sqrt3\le2)=\Phi(c)$, find c² (one decimal).

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Correct answer: 1.5 to 1.5

Explanation

Var(X₁²)=3−1=2. The CLT gives√n(Yn−1)⇒N(0,2), and the law of large numbers gives Yn→1. Slutsky’s theorem makes the displayed left random expression converge to1+N(0,2/3). Its probability of being≤2 isΦ(1/√(2/3))=Φ(√(3/2)). Hence c²=1.5.

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