GATE 2025 EC – Question 60
$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by:
$P(X=0,Y=0)=0.06$
$P(X=0,Y=1)=0.14$
$P(X=1,Y=0)=0.24$
$P(X=1,Y=1)=0.56$
The mutual information $I(X;Y)$ is ________ (rounded off to two decimal places).
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Show answer and explanation
Correct answer: -0.01 to 0.01
Explanation
The marginals are $P(X=0)=0.2$, $P(X=1)=0.8$, $P(Y=0)=0.3$, $P(Y=1)=0.7$. Every joint probability equals the product of the marginals ($0.2\times0.3=0.06$, $0.2\times0.7=0.14$, $0.8\times0.3=0.24$, $0.8\times0.7=0.56$), so $X$ and $Y$ are independent and $I(X;Y)=0$.