GATE 2022 ST – Question 27

Probability · Moments and Inequalities · 1 mark · Numerical answer

X_i are iid N(1,4),and S_n=ΣX_i². Find lim as n→∞ of [Var(S_n)/n−(E(S_n)/n)²],as an integer.

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Correct answer: 23 to 23

Explanation

For N(μ,σ²),E X²=σ²+μ²=5 and Var(X²)=2σ⁴+4μ²σ²=32+16=48. Independence gives E S_n=5n,Var S_n=48n. The expression is 48−25=**23** for every n.

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