Variance of Linear Combination of Random Variables

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Theorem

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

Let the variances of $X$ and $Y$ be finite.

Let $a$ and $b$ be real numbers.


Then the variance of $a X + b Y$ is given by:

$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}$

where $\cov {X, Y}$ is the covariance of $X$ and $Y$.


Corollary

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

Then the variance of $a X + b Y$ is given by:

$\var {a X + b Y} = a^2 \, \var X + b^2 \, \var Y$


Proof

\(\ds \var {a X + b Y}\) \(=\) \(\ds \expect {\paren {a X + b Y - \expect {a X + b Y} }^2}\) Definition of Variance
\(\ds \) \(=\) \(\ds \expect {\paren {a X + b Y - a \, \expect X - b \, \expect Y}^2}\) Expectation is Linear
\(\ds \) \(=\) \(\ds \expect {\paren {a \paren {X - \expect X} + b \paren {Y - \expect Y} }^2}\)
\(\ds \) \(=\) \(\ds \expect {a^2 \paren {X - \expect X}^2 + b^2 \paren {Y - \expect Y}^2 + 2 a b \paren {X - \expect X} \paren {Y - \expect Y} }\)
\(\ds \) \(=\) \(\ds a^2 \, \expect {\paren {X - \expect X}^2} + b^2 \, \expect {\paren {Y - \expect Y}^2} + 2 a b \, \expect {\paren {X - \expect X} \paren {Y - \expect Y} }\) Expectation is Linear
\(\ds \) \(=\) \(\ds a^2 \, \var X + b^2 \, \var Y + 2 a b \, \cov {X, Y}\) Definition of Variance, Definition of Covariance

$\blacksquare$