# Covariance of Multiples of Random Variables

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## 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 |

$\blacksquare$