# Covariance of Independent Random Variables is Zero

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## Theorem

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 $0$.

## Proof

\(\ds \cov {X, Y}\) | \(=\) | \(\ds \expect {X Y} - \expect X \expect Y\) | 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 | |||||||||||

\(\ds \) | \(=\) | \(\ds 0\) |

$\blacksquare$