Covariance of Independent Random Variables is Zero

From ProofWiki
Jump to navigation Jump to search


Let $X$ and $Y$ be independent random variables.

Let the expectations of $X$ and $Y$ exist and be finite.

Then the covariance of $X$ and $Y$ is $0$.


\(\ds \cov {X, Y}\) \(=\) \(\ds \expect {X Y} - \expect X \expect Y\) Covariance as Expectation of Product minus Product of Expectations
\(\ds \) \(=\) \(\ds \expect X \expect Y - \expect X \expect Y\) Condition for Independence from Product of Expectations
\(\ds \) \(=\) \(\ds 0\)