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GATE CSE 2024 Session 2 Q44: Bound on the mean of a product of [0,1] random variables

Question

Let xx and yy be random variables, not necessarily independent, that take real values in the interval [0,1][0, 1]. Let z=xyz = xy and let the mean values of x,y,zx, y, z be xˉ,yˉ,zˉ\bar{x}, \bar{y}, \bar{z}, respectively. Which one of the following statements is TRUE?

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Answer: D (official GATE 2024 final answer key)

Concept used: Without independence, E[xy]E[xy] can be above or below E[x]E[y]E[x]E[y]; only pointwise bounds survive.

First step: Since 0≤y≤10 \le y \le 1 and x≥0x \ge 0, xy≤xxy \le x for every outcome; take expectations.

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Source: official GATE 2024 Session 2 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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