#independence

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$X$ and $Y$ have joint density $p(x, y) = 4xy$ for $x, y \in [0, 1]$ and 0 elsewhere. What is $\textrm{Cov}(X, Y)$ ?

0, because the density factors as $(2x)(2y)$ , so $X$ and $Y$ are independent1
$4/9$ , the value of $E[XY] = \iint xy \cdot 4xy\,dx\,dy$ 2
Positive, because the density is largest at $(1, 1)$ , where both variables are large3
Nonzero, because independence would need a constant density on the square4

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