Covariance is Symmetric

Theorem

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

Suppose $\cov {X, Y}$ and $\cov {Y, X}$ exist.

Then $\cov {X, Y} = \cov {Y, X}$.

Proof

\(\ds \cov {X, Y}\)

\(=\)

\(\ds \expect {\paren {X - \expect X} \paren {Y - \expect Y} }\)

Definition of Covariance

\(\ds \)

\(=\)

\(\ds \expect {\paren {Y - \expect Y} \paren {X - \expect X} }\)

Real Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds \cov {Y, X}\)

Definition of Covariance

$\blacksquare$