Covariance as Expectation of Product minus Product of Expectations
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Theorem
Let $X$ and $Y$ be random variables.
Let the expectations of $X$ and $Y$ exist and be finite.
Then the covariance of $X$ and $Y$ is given by:
- $\cov {X, Y} = \expect {X Y} - \expect X \expect Y$
Proof
\(\ds \cov {X, Y}\) | \(=\) | \(\ds \expect {\paren {X - \expect X} \paren {Y - \expect Y} }\) | Definition of Covariance | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X Y - X \expect Y - Y \expect X + \expect X \expect Y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X Y} - \expect Y \expect X - \expect Y \expect X + \expect X \expect Y\) | Expectation is Linear, Expectation of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X Y} - \expect X \expect Y\) |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): covariance
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): covariance