Joint Distributions · Order Statistics · 2 marks · Numerical answer
Three iid observations have density 2x on (0,1),zero elsewhere. Let p be the probability the maximum is at least twice the minimum. Find p to three decimals.
Let U_i=X_i²,which are iid uniform. The event is U_max≥4U_min. The joint density of min u,max v for n=3 is 6(v−u),0<u<v<1. The complementary probability is ∫₀¹∫_(v/4)^v6(v−u)dudv=9/16. Hence p=1−9/16=7/16=.4375,rounded **0.438**.
More Order Statistics questions
GATE 2022 ST Q62– Y₁<⋯<Y n are order statistics of an iid sample from a continuous distribution symmetric…
GATE 2021 ST Q27– The kth order statistic from a size-eight Uniform(0,2) sample has density (7/32)x⁶(2−x)…
GATE 2021 ST Q48– X (1)<⋯<X (5) are order statistics of a size-five Uniform(0,θ) sample. P:3X (2) is…
GATE 2021 ST Q58– X n are iid Uniform(0,3). Independently,Y has PMF P(Y=k)=1/[(e−1)k!],k=1,2,… . Find…
GATE 2025 ST Q44– Independent X,Y₁,Y₂ have exponential rates 2,1,1 respectively. R i is the ascending rank…
GATE 2024 ST Q32– Three iid observations have density 2x on (0,1),zero otherwise. Find P(min(X₁,X₂,X₃)≥E X₁)…