GATE 2026 ST – Question 14

Probability · Random Variables and Transforms · 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?

P(XY=0)=1/6
P(XY=2)=1/6
E(X)=0
Var(Y)=1

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Show answer and explanation

Correct answer: (A) P(XY=0)=1/6

Explanation

Independence gives M_Y=M_(X+Y)/M_X=eᵗ, so Y=1 almost surely. X takes 0,−1,2 with probabilities 1/6,2/6,3/6. Thus P(XY=0)=1/6, P(XY=2)=1/2, E(X)=2/3 and Var(Y)=0. **A**.

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