X and Y are independent fair bits, each 0 or 1. A dataset keeps only the rows with X + Y ≥ 1. What is Cov(X, Y) within the kept rows? Give the answer as a fraction in lowest terms, with its sign.

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$X$ and $Y$ are independent fair bits, each $0$ or $1$ . A dataset keeps only the rows with $X + Y \geq 1$ . What is $Cov (X, Y)$ within the kept rows? Give the answer as a fraction in lowest terms, with its sign.

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Hint

List the rows that survive. They stay equally likely.

Solution

The kept rows are $(1, 0)$ , $(0, 1)$ and $(1, 1)$ , equally likely.
So $E [X] = E [Y] = \frac{2}{3}$ and $E [X Y] = \frac{1}{3}$ .
The covariance is $\frac{1}{3} - \frac{2}{3} \times \frac{2}{3} = - \frac{1}{9}$ .

Answer: $- \frac{1}{9}$

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