GATE 2023 ST – Question 14

Probability · Random Variables and Transforms · 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?

X is a random variable relative to 𝒢,but Y is not.
Y is a random variable relative to 𝒢,but X is not.
Neither is a random variable relative to 𝒢.
Both are random variables relative to 𝒢.

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Show answer and explanation

Correct answer: (A) X is a random variable relative to 𝒢,but Y is not.

Explanation

X is constant on each measurable atom,and its level-set preimages belong to 𝒢. In contrast,{Y≤2}=[0,1.5] does not belong to 𝒢. Thus only X is measurable. **A**.

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