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GATE 2025 EE – Question 15

Engineering Mathematics · Probability and Statistics: Random variables · 1 mark · Multiple choice

Consider discrete random variables $X$ and $Y$ with probabilities as follows:

$P(X=0\text{ and }Y=0)=\dfrac14$

$P(X=1\text{ and }Y=0)=\dfrac18$

$P(X=0\text{ and }Y=1)=\dfrac12$

$P(X=1\text{ and }Y=1)=\dfrac18$

Given $X=1$, the expected value of $Y$ is

  1. $\dfrac14$
  2. $\dfrac12$
  3. $\dfrac18$
  4. $\dfrac13$

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Correct answer: (B) $\dfrac12$

Explanation

$P(X=1)=\tfrac18+\tfrac18=\tfrac14$. $P(Y=1\mid X=1)=\dfrac{1/8}{1/4}=\dfrac12$, and since $Y$ takes values 0 and 1, $E[Y\mid X=1]=\dfrac12$.