GATE Joint Distributions: Joint and Conditional Distributions – Previous Year Questions

16 GATE previous year questions on Joint and Conditional Distributions (Joint Distributions, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q17 (1 mark, Multiple choice) – (X,Y) is bivariate normal with mean zero,unit marginal variances and correlation ρ∈(−1,1). Let Φ ρ(0,0)=P(X≤0,Y≤0). Find Kendall’s τ.
2. GATE 2021 ST Q44 (2 marks, Multiple choice) – The joint density is 4/(x+y)³ for x>1,y>1,zero elsewhere. Which claim is NOT true?
3. GATE 2021 ST Q45 (2 marks, Multiple choice) – X₁,X₂,X₃ are uncorrelated with common variance σ². Let Y₁=2X₁+X₂+X₃,Y₂=X₁+2X₂+X₃,Y₃=X₁+X₂+2X₃. P:the covariance-matrix eigenvalues of (Y₁,Y₂,Y₃) sum…
4. GATE 2022 ST Q41 (2 marks, Multiple choice) – The joint density of (X,Y,Z) is (2x+3y+z)/3 on the unit cube,zero elsewhere. Which point lies on the regression surface of X on (Y,Z)?
5. GATE 2022 ST Q64 (2 marks, Numerical answer) – X∼Uniform(0,4),and Y X=x∼Uniform(0,x²/4). Find E Y²,to three decimals.
6. GATE 2023 ST Q19 (1 mark, Multiple choice) – The joint density is f(x,y)=8xy for 0<x<y<1,zero otherwise. If E(X Y=y₀)=1/2,find y₀.
7. GATE 2023 ST Q47 (2 marks, Multiple choice) – The joint PMF is p(x,y)=c/2(x+y+2) for nonnegative integers x≠y,zero otherwise. Which holds?
8. GATE 2023 ST Q56 (2 marks, Multiple select) – The joint PMF is e⁻²/[x!(y−x)!] for integers 0≤x≤y,zero otherwise. Which hold?
9. GATE 2024 ST Q20 (1 mark, Multiple choice) – Which statement is not true for random variables X,Y?
10. GATE 2024 ST Q51 (2 marks, Multiple select) – (X,Y) has joint density 3/4 on −1≤x≤1,x²≤y<1,zero elsewhere. Which hold?
11. GATE 2025 ST Q15 (1 mark, Multiple choice) – (X₁,X₂,X₃) is multinomial with 100 trials and probabilities (3/10,1/10,3/5). Find E(X₂ X₃=40).
12. GATE 2025 ST Q41 (2 marks, Multiple choice) – For λ>0, joint masses are p (X,Y)(m,n)=λⁿe(−λ)/[2ⁿm!(n−m)!],m=0,…,n,n=0,1,…. Which holds?
13. GATE 2026 ST Q15 (1 mark, Multiple choice) – Identically distributed X,Y have variance σ²∈(0,∞). Their correlation coefficient is which expression?
14. GATE 2026 ST Q22 (1 mark, Multiple choice) – Let iid continuous pairs (X i,Y i) have joint CDF F (X,Y)(x,y)=F X(x)F Y(y). For distinct sample pairs i,j find P[(X i−X j)(Y i−Y j)>0].
15. GATE 2026 ST Q28 (1 mark, Numerical answer) – X,Y have joint density x+y on 0<x<1,0<y<1, zero otherwise. Find 72[Var(X)+Var(Y)] (integer).
16. GATE 2026 ST Q41 (2 marks, Multiple choice) – Independent iid geometric X,Y have P(X=x)=p(1−p)x,x=0,1,…,0<p<1. Which holds?