GATE Probability: Moments and Inequalities – Previous Year Questions

7 GATE previous year questions on Moments and Inequalities (Probability, Statistics) with answers and explanations, from every paper.

1. GATE 2021 ST Q11 (1 mark, Multiple choice) – X is a nonconstant positive random variable with E X=9. Which holds?
2. GATE 2021 ST Q32 (1 mark, Numerical answer) – X has density (3/13)(1−x)(9−x) on 0<x<1,zero otherwise. Find (4/3)E[X(X²−15X+27)],to two decimals.
3. GATE 2022 ST Q27 (1 mark, Numerical answer) – X i are iid N(1,4),and S n=ΣX i². Find lim as n→∞ of [Var(S n)/n−(E(S n)/n)²],as an integer.
4. GATE 2023 ST Q52 (2 marks, Multiple select) – X is positive and continuous with finite mean. Let Y=⌊X⌋. Which hold?
5. GATE 2024 ST Q39 (2 marks, Multiple choice) – X has density c₁/√x for 0<x≤1,c₂/x² for x>1,zero otherwise. If P(1/4≤X≤4)=5/8, consider I:P(3≤X≤5)=1/12. II:neither X nor 1/X has finite expectation.…
6. GATE 2026 ST Q17 (1 mark, Multiple choice) – X has density 4x²e(−2x) for x>0 and zero otherwise. If Y=ln X, which holds?
7. GATE 2026 ST Q52 (2 marks, Multiple select) – X has full support 1,2,3,…, E(X²)<∞ and CDF F. Which hold?