GATE 2021 ST – Question 59

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

X_n are iid Exp(rate 1),and X_(n)=max{X₁,…,X_n}. If X_(n)−ln n converges in distribution to Z,find the median of Z,to two decimals.

Show answer and explanation

Correct answer: 0.32 to 0.42

Explanation

For fixed z,P(X_(n)−ln n≤z)=[1−e^(−z)/n]^n→exp(−e^(−z)). The median m solves exp(−e^(−m))=1/2,so e^(−m)=ln 2 and m=−ln(ln 2)≈**0.37**.

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