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GATE 2023 ST – Question 52

Probability · Moments and Inequalities · 2 marks · Multiple select

X is positive and continuous with finite mean. Let Y=⌊X⌋. Which hold?

  1. P(Y≤u)≤P(X≤u) for every u≥0.
  2. P(Y≥u)≤P(X≥u) for every u≥0.
  3. E X<E Y.
  4. E X>E Y.

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Show answer and explanation

Correct answer: (B) P(Y≥u)≤P(X≥u) for every u≥0.; (D) E X>E Y.

Explanation

Since Y≤X,{Y≥u}⊂{X≥u},proving B. The CDF inequality goes in the reverse direction from A. Continuity makes P(X integer)=0,so X−Y>0 almost surely and it is bounded by 1. Its expectation is strictly positive,hence E X>E Y. **B,D**.

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