GATE 2024 DA – Question 65
Two fair coins are tossed independently. $X$ is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. $Y$ is a random variable that takes a value of 1 if at least one of the tosses is heads and 0 otherwise.
The value of the covariance of $X$ and $Y$ is ______ (rounded off to three decimal places).
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Correct answer: 0.053 to 0.073
Explanation
$E[X] = P(\text{HH}) = \frac{1}{4}$ and $E[Y] = P(\text{at least one head}) = \frac{3}{4}$. Since $X = 1$ means that both are heads, which makes $Y = 1$, we get $XY = X$ and $E[XY] = \frac{1}{4}$. The covariance is $E[XY] - E[X]E[Y] = \frac{1}{4} - \frac{3}{16} = \frac{1}{16} = 0.0625 \approx 0.063$.