GATE 2025 ST – Question 16

Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem · 1 mark · Multiple choice

The X_n are iid with density 1/[π(1+x²)] on ℝ. Define Y_n=1/2+arctan(X_n)/π. Which convergence in probability holds?

n⁻¹ΣY_i→1/2
n⁻¹ΣY_i→0
n⁻¹ΣX_i→0
n⁻¹ΣX_i→1/2

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) n⁻¹ΣY_i→1/2

Explanation

Y_n is the Cauchy CDF evaluated at X_n, hence iid uniform(0,1). The law of large numbers gives average Y→1/2. The average of iid standard Cauchy variables remains standard Cauchy for every n and cannot converge in probability to a constant. **A**.

More Laws of Large Numbers and Central Limit Theorem questions

GATE 2024 ST Q43 – The iid X n have density e(−(x−θ)),x≥θ,zero otherwise,θ>0. I:the sample average converges…
GATE 2023 ST Q57 – X n are iid with mean zero and variance one. Set Y n=(X₁X₂+X₃X₄+⋯+X (2n−1)X (2n))/n. Which…
GATE 2023 ST Q61 – 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…
GATE 2022 ST Q31 – 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…
GATE 2021 ST Q29 – X n are iid Uniform(0,2). Set Z n=−ln[(∏ (i=1)n(2−X i))(1/n)]. Find its almost-sure…
GATE 2021 ST Q59 – X n are iid Exp(rate 1),and X (n)=maxX₁,…,X n. If X (n)−ln n converges in distribution to…

All Laws of Large Numbers and Central Limit Theorem questions →