The GATE Grind

GATE 2025 DA – Question 11

Probability and Statistics · Conditional expectation and variance · 1 mark · Multiple choice

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

  1. $E[X|Y]$
  2. $E[X]E[Y]$
  3. $E[X]$
  4. $E[Y]$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) $E[X]$

Explanation

This is the law of total expectation (the tower rule): averaging the conditional mean of $X$ over all values of $Y$ gives the overall mean, $E[E[X \mid Y]] = E[X]$.