GATE Joint Distributions: Functions of Random Vectors – Previous Year Questions

5 GATE previous year questions on Functions of Random Vectors (Joint Distributions, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q42 (2 marks, Multiple choice) – X∼Uniform[−π/2,π/2]. Which statement is NOT true?
2. GATE 2023 ST Q53 (2 marks, Multiple select) – X has density e(−x) for x≥0,zero otherwise. Which hold? U(a,b) denotes the uniform law on (a,b).
3. GATE 2023 ST Q55 (2 marks, Multiple select) – U,V are iid with density λ²x e(−λx),x>0,λ>0. Which statements hold?
4. GATE 2024 ST Q61 (2 marks, Numerical answer) – The joint MGF is M (X,Y)(u,v)=exp(u²/2)/(1−2v)³ for real u and v<1/2. Find E(6X²/Y) to two decimals.
5. GATE 2026 ST Q63 (2 marks, Numerical answer) – Independent U₁,U₂,U₃ are exponential with respective rates 1,2,3. Let X=min(U₁,U₃),Y=min(U₂,U₃). Find P(X=Y) to two decimals.