Covariance of Linear Combination of Random Variables with Another

Theorem

Let $X, Y, Z$ be random variables.

Let $a, b$ be real numbers.

Then:

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

$\ds \cov {a X + b Y, Z}$

$=$

$\ds \expect {\paren {a X + b Y} Z} - \expect {a X + b Y} \expect Z$

$\ds $

$=$

$\ds a \expect {X Z} + b \expect {Y Z} - \paren {a \expect X + b \expect Y} \expect Z$

$\ds $

$=$

$\ds a \paren {\expect {X Z} - \expect X \expect Z} + b \paren {\expect {Y Z} - \expect Y \expect Z}$

$\ds $

$=$

$\ds a \cov {X, Z} + b \cov {Y, Z}$

$\blacksquare$