GATE Convergence of Random Variables: Laws of Large Numbers and Central Limit Theorem – Previous Year Questions
8 GATE previous year questions on Laws of Large Numbers and Central Limit Theorem (Convergence of Random Variables, Statistics) with answers and explanations, from every paper.
- 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 limit,to two decimals.
- 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 Z,find the median of Z,to two decimals.
- 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 √n[X̄ n(3−X̄ n)−20/9]→N(0,σ²) in distribution,find σ² to two…
- 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 hold?
- 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 n)−e⁻¹] converges in distribution to N(0,σ²),find σ² to two…
- GATE 2024 ST Q43 – The iid X n have density e(−(x−θ)),x≥θ,zero otherwise,θ>0. I:the sample average converges in probability to (θ+1)/2. II:lim E[min(X₁,…,X n)]=θ. Which…
- GATE 2025 ST Q16 – 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?
- GATE 2026 ST Q54 – Independent X (2k−1)~Bin(1,θ),X (2k)~Bin(1,1−θ),0<θ<1. The Y k are iid Poisson(λ),λ>0. Put S 2n= k=1n(X 2k-1-X 2k+1-2), W n= k=1nY k2 and σ…