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).
Answer
0.06
Explanation
Use \(\operatorname{Cov}(X, Y) = E[XY] - E[X]\,E[Y]\). The four outcomes HH, HT, TH, TT are equally likely.
- \(X = 1\) only for HH, so \(E[X] = \frac{1}{4}\).
- \(Y = 1\) for every outcome except TT, so \(E[Y] = \frac{3}{4}\).
- Whenever \(X = 1\), also \(Y = 1\), so \(XY = X\) and \(E[XY] = \frac{1}{4}\).
\[ \operatorname{Cov}(X, Y) = \frac{1}{4} - \frac{1}{4} \cdot \frac{3}{4} = \frac{1}{16} = 0.0625 \]
The covariance is positive because two heads guarantee at least one head. As 0.0625 lies exactly halfway, the key accepts both 0.062 and 0.063.
Answer: 0.063