Is $E(xy) \geq E(x) E(y)$?

Question

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

score 5 · Accepted Answer · answered Jun 21 '13 at 01:50

5

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.

answered Jun 21 '13 at 01:50

Tunococ

Good point, thanks. But regular variance is always nonnegative right? – badatmath Jun 21 '13 at 01:54
1

@badatmath Yes. The regular variance is always non-negative because it is an expectation of a non-negative quantity. – Tunococ Jun 21 '13 at 02:01
Oh right, because it can be written as $E((X - E(X))^2)$. Gotcha. – badatmath Jun 21 '13 at 02:03

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