Covariance of Multiples of Random Variables

Theorem

Let $X, Y$ be random variables.

Let $a, b$ be real numbers.

Then:

$\cov {a X, b Y} = a b \cov {X, Y}$

Proof

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

\(=\)

\(\ds \expect {a X b Y} - \expect {a X} \expect {b Y}\)

Covariance as Expectation of Product minus Product of Expectations

\(\ds \)

\(=\)

\(\ds a b \expect {X Y} - a b \expect X \expect Y\)

Expectation of Linear Transformation of Random Variable

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Covariance as Expectation of Product minus Product of Expectations

$\blacksquare$