GATE Probability: Random Variables and Transforms – Previous Year Questions

17 GATE previous year questions on Random Variables and Transforms (Probability, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q26 (1 mark, Numerical answer) – 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 A=⋃ (n≥1)A n. Given P(X≤1)=1/2 and E X=5/3,find P(X∈A),to two…
2. GATE 2021 ST Q57 (2 marks, Numerical answer) – 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 two decimals.
3. GATE 2022 ST Q29 (1 mark, Numerical answer) – X has density c(x−⌊x⌋) on 0<x<3,zero elsewhere. Find P(X∈[1/2,2]),to two decimals.
4. GATE 2022 ST Q30 (1 mark, Numerical answer) – X has MGF M(t),mean 1/2 and variance 1/4. Y has MGF H(t)=[(3/4)e(2t)+1/4]M(t) near zero. Find Var Y as an integer.
5. GATE 2022 ST Q44 (2 marks, Multiple choice) – 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…
6. GATE 2022 ST Q49 (2 marks, Multiple select) – For a>0,suppose P(aX/(2π)∈ℤ)=1. Let φ X(t) be its characteristic function. Which hold?
7. GATE 2023 ST Q14 (1 mark, Multiple choice) – On Ω=[0,2], let 𝒢=∅,Ω,[0,1],(1,2]. Define X=1 on [0,1] and 2 on (1,2]; Y=2 on [0,1.5] and 3 on (1.5,2]. Which statement holds?
8. GATE 2023 ST Q15 (1 mark, Multiple choice) – Let Φ be the standard normal CDF. X has CDF F(x)=Φ(x) for x<−1 and Φ(x+1) for x≥−1. Which holds?
9. GATE 2023 ST Q46 (2 marks, Multiple choice) – X has CDF 0 for x<−1,(x+1)/4 for −1≤x<0,(x+3)/4 for 0≤x<1,and 1 for x≥1. Which holds?
10. GATE 2024 ST Q17 (1 mark, Multiple choice) – X takes only values 1,2 and E X=10/7. Find the fourth derivative of its MGF at zero.
11. GATE 2024 ST Q38 (2 marks, Multiple choice) – X has density (3/8)(x+1)² on −1<x<1,zero otherwise. For Y=1−X², find P(Y≤3/4).
12. GATE 2024 ST Q50 (2 marks, Multiple select) – On Ω=1,2,…, let P(k)=2(−k) and X(ω)=ω. Which hold?
13. GATE 2025 ST Q28 (1 mark, Numerical answer) – X has CDF F with lim (h→0−)F(3+h)=1/4 and F(3)=3/4. Find 16P(X=3) (integer).
14. GATE 2025 ST Q39 (2 marks, Multiple choice) – X has CDF 0 for x<0, α(1+2x²) for 0≤x<1 and 1 for x≥1. Its median is 1/√2. Find α.
15. GATE 2025 ST Q58 (2 marks, Numerical answer) – Independent X,Y,Z have MGFs M X(t)=(2+eᵗ)²/9,M Y(t)=exp(eᵗ−1),M Z(t)=exp[2(eᵗ−1)]. Find 10P(X>Y+Z) to two decimals.
16. GATE 2026 ST Q14 (1 mark, Multiple choice) – Independent discrete X,Y have M X(t)=(1+2e(−t)+3e(2t))/6 and M (X+Y)(t)=(2+eᵗ+3e(3t))/6. Which holds?
17. GATE 2026 ST Q30 (1 mark, Numerical answer) – Continuous X has CDF F. Let Y=−3ln F(X). Find E(Y) (integer).