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GATE DA 2024 Q59: Conditional expectation from a joint density (marks to all)

Question

Consider a joint probability density function of two random variables XX and YY

fX,Y(x,y)={2xy,0<x<2, 0<y<x0,otherwisef_{X,Y}(x, y) = \begin{cases} 2xy, & 0 < x < 2,\ 0 < y < x \\ 0, & \text{otherwise} \end{cases}

Then, E[Y∣X=1.5]E[Y \mid X = 1.5] is ______.

The official answer key gives marks to all for this question, so there is no answer to check. It is shown for practice and discussion.

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Answer: Marks to all (official GATE 2024 final answer key)

Concept used: A joint density must integrate to 1 over its support; conditional densities come from fY∣X(y∣x)=fX,Y(x,y)/fX(x)f_{Y \mid X}(y \mid x) = f_{X,Y}(x, y)/f_X(x).

First step: Check the total probability first: ∫02∫0x2xy dy dx=∫02x3 dx=4≠1\int_0^2 \int_0^x 2xy \, dy \, dx = \int_0^2 x^3 \, dx = 4 \ne 1, so the given function is not a valid density. The official key gives marks to all.

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

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