I:For a variable with density e^(−|x|)/2 on ℝ,there exist iid X₁,X₂ such that X₁−X₂ has this law. II:For a variable uniform on (−2,2),there exist iid Y₁,Y₂ such that Y₁−Y₂ has this law. Which hold?
For I,take iid Exp(1) variables;their difference has the stated Laplace density. The characteristic function of any iid difference is φ(t)φ(−t)=|φ(t)|²≥0. Uniform(−2,2) has characteristic function sin(2t)/(2t),which takes negative values,so II is impossible. **A**.
More Random Variables and Transforms questions
GATE 2022 ST Q49– For a>0,suppose P(aX/(2π)∈ℤ)=1. Let φ X(t) be its characteristic function. Which hold?
GATE 2021 ST Q26– X has CDF 0 for x<1,a/2 for 1≤x<2,c/6 for 2≤x<3,and 1 for x≥3. Let A n=[1+1/n,3−1/n] and…
GATE 2021 ST Q57– X has MGF M(t)=(et−1)/[t(1−t)] for t<1,with the continuous value at t=0. Find P(X>1),to…
GATE 2026 ST Q14– Independent discrete X,Y have M X(t)=(1+2e(−t)+3e(2t))/6 and M (X+Y)(t)=(2+eᵗ+3e(3t))/6.…
GATE 2026 ST Q30– Continuous X has CDF F. Let Y=−3ln F(X). Find E(Y) (integer).
GATE 2025 ST Q28– X has CDF F with lim (h→0−)F(3+h)=1/4 and F(3)=3/4. Find 16P(X=3) (integer).