GATE DA 2025 Q11: Law of total expectation: E[E[X|Y]]

Suppose $X$ and $Y$ are random variables. The conditional expectation of $X$ given $Y$ is denoted by $E[X|Y]$ . Then $E[E[X|Y]]$ equals

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Answer: C (official GATE 2025 final answer key)

Concept used: Law of total expectation (tower property).

First step: $E[X|Y]$ is a function of $Y$ ; averaging it over all values of $Y$ recovers the overall average of $X$ .

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key