Is $E(xy) \geq E(x) E(y)$? Is $E(xy) \geq E(x) E(y)$ always?  Where $E$ means expectation.
Intuitively I feel like covariance should always be nonnegative, but I cannot prove why.
Thanks for your help

That is not true. For example, let $x$ be $0$ or $1$ with equal probability of $0.5$, and let $y = 1 - x$. Then $E(xy) = 0$, but $E(x)E(y) = 0.25$.
About the second sentence, it is also false. For example, let $y = -x$. Then the covariance will be negative if $x$ has non-zero variance.
